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MR2720543 83Cxx (81T20 81V17)

Vacaru, Sergiu I. (R-IASI-SC)Two-connection renormalization and non-holonomic gauge models of Einstein gravity.(English summary)Int. J. Geom. Methods Mod. Phys.7 (2010),no. 5,713–744.

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MR2657560 53B40 (83D05)

Vacaru, Sergiu I. (R-IASI-SC)Critical remarks on Finsler modifications of gravity and cosmology by Zhe Chang and XinLi [ MR2463269; MR2536194]. (English summary)Phys. Lett. B690(2010),no. 3,224–228.

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MR2608569 83E05Vacaru, Sergiu I. (R-IASI-SC)On general solutions for field equations in Einstein and higher dimension gravity. (Englishsummary)Internat. J. Theoret. Phys.49 (2010),no. 4,884–913.

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Vacaru, Sergiu I. (R-IASI-SC)Einstein gravity in almost Kahler and Lagrange-Finsler variables and deformationquantization. (English summary)J. Geom. Phys.60 (2010),no. 10,1289–1305.

{A review for this item is in process.}

References

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2. S. Vacaru, Generalized Lagrange transforms: Finsler geometry methods and deformation quan-tization of gravity, An. Stiint. Univ. Al. I. Cuza lasi. Mat. (NS) LIII (Suppl.) (2007) 327–342.MR2522405 (2010i:53178)

3. S. Vacaru, Deformation quantization of nonholonomic almost Kahler models and Einsteingravity, Phys. Lett. A 372 (2008) 2949–2955.MR2404752 (2009g:53137)

4. S. Vacaru, Branes and quantization of an a—model complexification for Einstein gravity inalmost Kahler variables, Int. J. Geom. Methods Mod. Phys. 6 (2009) 873–909.MR2571415

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sical Quantum Gravity 21 (2004) R53.MR2079936 (2005g:83043)11. T. Thiemann, Modern Canonical Quantum General Relativity, Cambrigde University Press,

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(99h:81184)16. C. Misner, K. Thorne, J. Wheeler, Gravitation, Freeman, 1973.MR0418833 (54 #6869)17. F.A. Berezin, General concept of quantization, Comm. Math. Phys. 40 (1975) 153–174.

MR0411452 (53 #15186)18. F. Bayen. M. Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, Deformation theory

and quantization. I. Deformations of symplectic structures, Ann. Phys. 111 (1978) 61–110.MR0496157 (58 #14737a)

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20. H. Garcia-Compean, J. Plebanski, M. Przanowski, The geometry of deformation quantizationand self-dual gravity, in: Jerusalem 1997. Recent Develompents in Theoretical and Experimen-tal General Relativity. Gravitation, and Relativistic Field Theories, Pt. A Marcel Grossman.Hebrew University Press, Jerusalem, 1997, pp. 442–444.MR1891913 (2003b:83081)

21. F. Antonsen, Deformation quantization of gravity. arXiv:gr-qc/9712012.22. H. Quevedo, J. Tafoya, Towards the deformation quantization of linearized gravity, Gen. Rela-

tivity Gravitation 37 (2005) 2083–2092.MR2193045 (2007b:83071)23. C. Castro, W-geometry from Fedosov’s deformation quantization, J. Geom.p phys. 33 (2000)

173–190.MR1738151 (2002b:53140)24. C. Castro, Moyal deformations of gravity viaSU(∞) gauge theories, branes and topological

Chern-Simons matrix models, Gen. Relativity Gravitation 36 (2004) 2605–2635.MR2109612(2005j:83072)

25. Y. Cho, K. Soh, J. Yoon, W. Park, Gravitation as gauge theory of diffeomorphysm group, Phys.Lett. B 286 (1992) 252–255.MR1175136 (93i:83082)

26. C. Castro, String and membranes from Einstein gravity, matrix models andW∞ gauge theories,Invert. J. Sci. Technol. 2 (2008) 84–92.

27. R. Miron, M. Anastasiei, Vector Bundles and Lagrange Spaces with Applications to Relativity,Geometry Balkan Press, Bukharest, 1997, Translation from Romanian of (Editura AcademieiRomane, 1987).MR1469973 (98g:53002)

28. R. Miron, M. Anastasiei, The Geometry of Lagrange Spaces: Theory and Applications.FTPH, vol. 59. Kluwer Academic Publishers, Dordrecht, Boston, London, 1994.MR1281613(95f:53120)

29. S. Vacaru, P. Stavrinos. E. Gaburov, D. Gonta, Clifford and Riemann-Finsler structures in geo-metric mechanics and gravity, selected works, in: Differential Geometry-Dynamical Systems,Monograph, vol. 7, Geometry Balkan Press, 2006, www.mathem.pub.ro/dgds/mono/va-t.pdf.MR2255045 (2008i:53107)

30. S. Vacaru, Finsler and Lagrange geometries in Einstein and string gravity, Int. J. Geom. MethodsMod. Phys. 5 (2008) 473–511.MR2428807 (2009f:53027)

31. M. Anastasiei, S. Vacaru, Fedosov quantization of Lagrange-Finsler and Hamilton-Cartanspaces and Einstein gravity lifts on (co) tangent bundles, J. Math. Phys. 50 (2009) 013510.MR2492620 (2010f:53163)

32. F. Etayo, R. Santamarıa, S. Vacaru, Lagrange-Fedosov nonholonomic manifolds, J. Math. Phys.

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46 (2005) 032901.MR2125569 (2006a:53025)33. A.V. Karabegov, M. Schlichenmaier, Almost Kahler deformation quantization, Lett. Math.

Phys. 57 (2001) 135–148.MR1856906 (2002f:53156)34. M. Matsumoto, Foundations of Finsler geometry and special publishers. Dordrecht, Finsler

Spaces, Kaisisha, Shigaken, 1986.35. V.A. Oproiu, A Riemannian structure in Lagrange geometry, Rend. Sem. Fac. Sci. Univ.

Cagliari 55 (1985) 1–20.MR0860191 (87k:53049)36. P.O. Kazinski, S.L. Lyakhovich, A.A. Sharapov, Largange structure and quantization, J. High

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Energy Phys. 0503 (2005) 011.MR2151981 (2006a:81155)Note: This list reflects references listed in the original paper as accurately as possible with no

attempt to correct errors.

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MR2639102 (Review) 53B40 (53B50 83C57 83C65)

Vacaru, Sergiu I. (R-IASI-C)Finsler black holes induced by noncommutative anholonomic distributions in Einsteingravity. (English summary)Classical Quantum Gravity27 (2010),no. 10,105003, 19pp.

Summary: “We study Finsler black holes induced from Einstein gravity as possible effects of quan-tum spacetime noncommutativity. Such Finsler models are defined by nonholonomic frames noton tangent bundles but on (pseudo)Riemannian manifolds being compatible with standard theoriesof physics. We focus on noncommutative deformations of Schwarzschild metrics into locally an-isotropic stationary ones with spherical/rotoid symmetry. The conditions are derived when blackhole configurations can be extracted from two classes of exact solutions depending on noncom-mutative parameters. The first class of metrics is defined by nonholonomic deformations of thegravitational vacuum by noncommutative geometry. The second class of such solutions is inducedby noncommutative matter fields and/or effective polarizations of cosmological constants.”

Reviewed byLorenzo Sindoni

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MR2646782 (Review) 58A30 (53C80 83C05)

Vacaru, Sergiu I. (R-IASI-SC)Nonholonomic distributions and gauge models of Einstein gravity. (English summary)Int. J. Geom. Methods Mod. Phys.7 (2010),no. 2,215–246.

A new and interesting approach to gauge gravity is presented. It is based on the constructionof a gauge-like theory in certain bundle spaces endowed with nonholonomic distributions. Thelatter rely on a(2 + 2)-splitting of the base spacetime and the associated nonholonomic framestructure. In this context, general relativity theory is not generalized but rather reformulated incertain nonholonomic variables lifted to bundles of affine/de Sitter frames. To this purpose, theformalism of nonlinear connections and the methodology of nonholonomic frames are utilized.

In particular, after the presentation of an outline of the geometry of nonholonomic distributionson four-dimensional (pseudo)Riemannian manifolds, general relativity theory is reformulated innonholonomic variables for different linear connection structures. It is explained how all geomet-ric and physical information encoded in general relativity can be equivalently transformed intogeometric objects and constructions for any nonholonomic variables. Finally, using a relevantsplitting of connections, two models of gauge gravity in nonholonomic affine and de Sitter framebundles are constructed, with Yang-Mills–like equations equivalently reinterpreted as Einstein’sfield equations on the base spacetime. One should point out that the nontrivial torsion structurecontained in the equations obtained that are equivalent to Einstein’s field equations is inducednonholonomically by certain off-diagonal metric coefficients.

Reviewed byTheophanes Grammenos

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MR2601644 (Review) 53C21 (35Q53 35R01 37K10 37K25)

Vacaru, Sergiu I. (3-FLDS2)Curve flows and solitonic hierarchies generated by Einstein metrics. (English summary)Acta Appl. Math.110(2010),no. 1,73–107.

This work comes as a new addition to an interesting series of papers published by Vacaru onthe so-called nonholonomic frame method. The present paper’s aim is “to prove that respectivecurve flow solitonic hierarchies are generated by any (semi)Riemannian metricgαβ on a manifoldV of dimensionn + m, for n ≥ 2 andm ≥ 1, if such a space is endowed with a nonholonomic

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distribution defining ann +m spacetime splitting”.Initially, it is proven that, for any (semi)Riemannian metric on a nonholonomic manifold en-

dowed with a nonlinear connection structure and defining a spacetime splitting of dimensionn +m with n ≥ 2 andm ≥ 1, one can construct a metric-compatible linear connection with constantmatrix coefficients of curvature, in other words it is possible to construct a curvature tensor withconstant coefficients, calculated with respect to “nonlinear connection-adapted” frames. Thus, byapplying the method developed, it is always possible to convert data for known solutions of Ein-stein’s field equations into terms of constant curvature coefficients, in other words, into alternativeholonomic structures and metric compatible linear connections defined by the given metric tensor.Then, the geometry of curve flows adapted to a nonholonomic structure on a (semi)Riemannianmanifold is examined. A class of nonholonomic Klein spaces is considered for which the bi-Hamiltonian operators are obtained for a linear connection adapted to the nonlinear connectionstructure, for which the curvature coefficients are constant. The formalism of bi-Hamiltonian andvector solitonic structures is presented for arbitrary (semi)Riemannian spaces. In particular, thebasic equations for nonholonomic curve flows are introduced and the properties of Hamiltoniansymplectic and co-symplectic operators adapted to the nonlinear connection structure are given. Fi-nally, solitonic (of the mKdV and sine-Gordon type) hierarchies of bi-Hamiltonian nonholonomiccurve flows are constructed by wave map equations and recursion operators associated to the hor-izontal and vertical curve flows. In fact, the main theorem of the present paper, namely that thegeometric data on a (semi)Riemannian manifold naturally define a nonholonomic frame-adaptedbi-Hamiltonian flow hierarchy inducing nonholonomic solitonic configurations or, in other words,that any metric structure on a (semi)Riemannian manifold can be decomposed into solitonic datawith corresponding hierarchies of nonlinear waves, is rigorously proven.

Reviewed byTheophanes Grammenos

References

1. Vacaru, S.: Exact solutions with noncommutative symmetries in Einstein and gauge gravity. J.Math. Phys.46,042503 (2005)MR2131241 (2006b:83128)

2. Vacaru, S.: Parametric nonholonomic frame transforms and exact solutions in gravity. Int. J.Geom. Methods Mod. Phys. [IJGMMP]4, 1285–1334 (2007)MR2378384 (2008k:83030)

3. Vacaru, S.: Nonholonomic Ricci flows: II. Evolution equations and dynamics. J. Math. Phys.49,043504 (2008)MR2412299 (2009d:53091)

4. Vacaru, S.: Nonholonomic Ricci flows: III. Curve flows and solitonic hierarchies.arXiv:0704.2062 [math.DG]

5. Vacaru, S.: Nonholonomic Ricci flows, exact solutions in gravity, and symmetric andnonsymmetric metrics. Int. J. Theor. Phys.47 (2008). doi:10.1007/s10773–008-9841–8.arXiv:0806.3812 [gr-qc]MR2485601 (2010b:53124)

6. Miron, R., Anastasiei, M.: The Geometry of Lagrange Spaces: Theory and Applications. FTPH,vol. 59. Kluwer Academic, Dordrecht (1994)MR1281613 (95f:53120)

7. Vacaru, S.: Finsler and Lagrange geometries in Einstein and string gravity. Int. J. Geom.Methods. Mod. Phys. [IJGMMP]5, 473–511 (2008)MR2428807 (2009f:53027)

8. Vacaru, S., Stavrinos, P., Gaburov, E., Gonta, D.: Clifford and Riemann–Finsler Struc-

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tures in Geometric Mechanics and Gravity. Selected Works, Differential Geometry—Dynamical Systems, Monograph. 7. Geometry Balkan Press, Bucharest (2006).www.mathem.pub.ro/dgds/mono/va-t.pdf and gr-qc/0508023MR2255045 (2008i:53107)

9. Chou, K.-S., Qu, C.: Integrable equations arising from motions of plane curves. Phys. D162,9–33 (2002)MR1882237 (2003c:37106)

10. Mari Beffa, G., Sanders, J., Wang, J.-P.: Integrable systems in three-dimensional Riemanniangeometry. J. Nonlinear. Sci.12,143–167 (2002)MR1894465 (2003f:37137)

11. Athorne, C.: Local Hamiltonian structures of multicomponentKdV equations. J. Phys. A:Math. Gen.21,4549–4556 (1988)MR0974701 (90b:58061)

12. Sanders, J., Wang, J.-P.: Integrable systems inn dimensional Riemannian geometry. Mosc.Math. J.3, 1369–1393 (2003)MR2058803 (2004m:37142)

13. Sergyeyev, A.: Why nonlocal recursion operators produce local symmetries: new results andapplications. J. Phys. A: Math. Gen.38,3397–3407 (2005)MR2132718 (2005m:37166)

14. Foursov, M.V.: Classification of certain integrable coupled potentialKdV and modifiedKdV-type equations. J. Math. Phys.41,6173–6185 (2000)MR1779638 (2001j:37121)

15. Wang, J.-P.: Generalized Hasimoto transformation and vector sine-Gordon equation. In:Abenda, S., Gaeta, G., Walcher, S. (eds.) SPT 2002: Symmetry and Perturbation Theory (CalaGonone), pp. 276–283. World Scientific, River Edge (2002)MR1976678 (2004c:37172)

16. Anco, S.C.: Hamiltonian flows of curves inG/SO(n) and vector soliton equations ofmKdVand sine-Gordon type. Symmetry Integrability Geom: Methods Appl. (SIGMA)2, 044 (2006)MR2217753 (2007a:37073)

17. Anco, S.C.: Bi-Hamiltonian operators, integrable flows of curves using moving frames,and geometric map equations. J. Phys. A: Math. Gen.39, 2043–2072 (2006)MR2211976(2007g:37051)

18. Vacaru, S.: Spectral functionals, nonholonomic Dirac operators, and noncommutative Ricciflows. arXiv:0806.3814 [math-ph]MR2548633

19. Vacaru, S.: Deformation quantization of nonholonomic almost Kahler models and Einsteingravity. Phys. Lett. A372,2949–2955 (2008)MR2404752 (2009g:53137)

20. Bejancu, A.: Finsler Geometry and Applications. Ellis Horwood, Chichester (1990)MR1071171 (91i:53075)

21. Bao, D., Chern, S.-S., Shen, Z.: An Introduction to Riemann–Finsler Geometry. Graduate Textsin Math., vol. 200. Springer, Berlin (2000)MR1747675 (2001g:53130)

22. Vranceanu, G.: Sur les espaces non holonomes. C. R. Acad. Paris103,852–854 (1926)23. Vranceanu, G.: Lecons de geometrie differentielle, vol. II. Edition de l’Academie de la Re-

publique Populaire de Roumanie, Bucharest (1957)MR0124823 (23 #A2133)24. Horak, Z.: Sur les systemes non holonomes. Bull. Int. Acad. Sci. Boheme 1–18 (1927)25. Bejancu, A., Farran, H.R.: Foliations and Geometric Structures. Springer, New York (2005)

MR2190039 (2006j:53034)26. Vacaru, S.: Locally anisotropic kinetic processes and thermodynamics in curved spaces. Ann.

Phys. (N.Y.)290,83–123 (2001)MR1834616 (2002d:82064)27. Vacaru, S.: Stochastic processes and thermodynamics on curved spaces. Ann. Phys. (Leipzig)

9, 175–176 (2000). Special Issue

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28. Anco, S., Vacaru, S.: Curve flows in Lagrange–Finsler geometry, bi-Hamiltonian structuresand solitons. J. Geom. Phys. (2008). doi:10.1016/j.geomphys.2008.10.006. math-ph/0609070MR2378453 (2010a:37128)

29. Vacaru, S.: The entropy of Lagrange–Finsler spaces and Ricci flows. Rep. Math. Phys. (2008,accepted). math.DG/0701621MR2514350 (2010d:53073)

30. Vacaru, S.: Loop quantum gravity in Ashtekar and Lagrange-Finsler variables and Fedosovquantization of general relativity. arXiv:0801.4942 [gr-qc]

31. Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Am. Math. Soc.,Providence (2001)MR1834454 (2002b:53081)

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33. Sharpe, R.W.: Differential Geometry. Springer, New York (1997)MR1453120 (98m:53033)Note: This list reflects references listed in the original paper as accurately as possible with no

attempt to correct errors.

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MR2571415 (2011b:83050)83C45 (53C80 81S10)

Vacaru, Sergiu I. (R-IASI-SC)Branes and quantization for an A-model complexification of Einstein gravity in almostKahler variables. (English summary)Int. J. Geom. Methods Mod. Phys.6 (2009),no. 6,873–909.

It is known that deformation/geometric quantization does not constitute a standard quantizationprocedure for a number of reasons. Among other things, it does not lead to a natural Hilbert spaceon which the deformed algebra acts. However, there are several results obtained in this way thatcontribute to a deeper understanding of the quantization procedure in general. For example, defor-mation quantization of relativistic particles yields the same results as the canonical quantizationand path integral methods. In this work, characteristic of the author’s rigorous presentations, anew perspective regarding the quantization of Einstein’s gravity based on two-dimensional sigma-models is proposed, following the A-model quantization via branes (see the excellent work of S.Gukov and E. Witten [“Branes and quantization”, preprint, arxiv.org/abs/0809.0305]), and aimingat a closer look toward a systematic theory of quantum gravity in symplectic variables connectedwith an almost Kahler formulation of general relativity.

The paper starts with an introduction to the almost Kahler model of general relativity (withdetails given in an appendix). Here, nonholonomic distributions generated by regular pseudo-Lagrangians, nonlinear connections (canonical symplectic connections as distortions of the Levi-Civita connection), and almost Kahler variables are presented. Next, the quantization method for

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the A-model with nonholonomic branes is formulated. The necessary notions of the canonicalco-isotropic nonholonomic brane and the space of nonholonomic strings are thus developed forthe aforesaid formulation. In this context, it is shown that the Gukov-Witten method allows theconstruction of a physical, viable Hilbert space with a Hermitian inner product, for almost Kahlerquantum models of general relativity. Consequently, based also on some previous results ondeformation and loop quantum gravity stemming from the geometry of nonholonomic manifoldsand noncommutative spaces, the author succeeds in applying this method to explicitly constructquantum physical states for such a model.

Keeping in mind that different methods of nonlinear field theory quantization yield, in general,different quantum theories, the significance of working with almost Kahler variables, nonlin-ear connections and nonholonomic methods arises mainly from its prospective applicability tosupergravity/superstring theories and to complicated second-class constraint systems as well.

Reviewed byTheophanes Grammenos

c© Copyright American Mathematical Society 2011

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Citations

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MR2519980 (2010j:83129)83D05 (83C65)

Vacaru, Sergiu I. (3-FLDS2)Einstein gravity in almost Kahler variables and stability of gravity with nonholonomicdistributions and nonsymmetric metrics. (English summary)Internat. J. Theoret. Phys.48 (2009),no. 7,1973–1999.

Summary: “We argue that the Einstein gravity theory can be reformulated in almost Kahler (non-symmetric) variables with effective symplectic form and compatible linear connection uniquelydefined by a (pseudo) Riemannian metric. A class of nonsymmetric theories of gravitation on man-ifolds enabled with nonholonomic distributions is considered. We prove that, for certain types ofnonholonomic constraints, there are modelled effective Lagrangians which do not develop insta-bilities. It is also elaborated a linearization formalism for anholonomic noncommutative gravitytheories models and analyzed the stability of stationary ellipsoidal solutions defining some non-holonomic and/or nonsymmetric deformations of the Schwarzschild metric. We show how toconstruct nonholonomic distributions which remove instabilities in nonsymmetric gravity theo-ries. It is concluded that instabilities do not consist a general feature of theories of gravity withnonsymmetric metrics but a particular property of some models and/or unconstrained solutions.”

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MR2548633 (2011b:58062)58J42(46L87 53C44 58B34)

Vacaru, Sergiu I. (R-IASI-C)Spectral functionals, nonholonomic Dirac operators, and noncommutative Ricci flows.(English summary)J. Math. Phys.50 (2009),no. 7,073503, 24pp.

In this paper, a nonholonomic analogue of the theory of Spin manifolds and Dirac operators,together with the theory of spectral triples, is used to study the Ricci flow onN -anholonomicmanifolds and, in particular, to obtain Perelman’s functionals via the spectral action of noncom-mutative Riemannian geometry [A. H. Chamseddine and A. Connes, Comm. Math. Phys.186(1997), no. 3, 731–750;MR1463819 (99c:58010)].

First, the theory ofN -anholonomic manifolds is briefly recalled and suitable notions ofN -anholonomic spin structure, spin connection, and Dirac operator are defined by analogy with theunconstrained case. A special class of spectral triple, called “spectrald-triples”, is defined, and ananalogue of a theorem of A. Connes [Comm. Math. Phys.182(1996), no. 1, 155–176;MR1441908(98f:58024)] concerning commutative spectral triples is proved for commutative spectrald-triples.With this theory in place, the Ricci flow on anN -anholonomic manifold is realized by a one-parameter family of spectrald-triples, and Perelman’s first and second functionals, as well as anonholonomic average energy and entropy, are thus obtained from the asymptotic computation ofthe spectral action on that family of spectrald-triples.

Reviewed byBranimir Cacic

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Quantum Grav.19, 4029 (2002).MR1921411 (2003k:83085)26. P. Aschieri, C. Bolhmann, M. Dimitrijevic, F. Meyer, P. Suchpp, and J. Wess, Class. Quantum

Grav.22, 3511 (2005).27. R. J. Szabo, Class. Quantum Grav.23, R199 (2006).MR2271996 (2007i:81265)28. A. Zerjak and B. Dragovich, e-print arXiv:0708.3950.29. A. H. Chamseddine and A. Connes, Phys. Rev. Lett.99, 071302 (2007).MR2338507

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Berlin, 1997).MR1482228 (99b:58028)32. J. Madore,An Introduction to Noncommutative Differential Geometry and Its Physical Ap-

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Athens, 2002).

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1 and 2, edited by P. Deligne, D. S. Freed, P. I. Etingof, D. Kazhdan, and L. C. Jeffrey (AmericanMathematical Society, Providence, 1994).

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Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

c© Copyright American Mathematical Society 2011

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MR2514350 (2010d:53073)53C44 (53C60 70G45)

Vacaru, Sergiu I. (3-FLDS2)The entropy of Lagrange-Finsler spaces and Ricci flows. (English summary)Rep. Math. Phys.63 (2009),no. 1,95–110.

One of the main ideas of geometric mechanics is the investigation and exploitation of possi-ble (Lie) symmetries of physical systems. On the other hand, any Lagrange mechanics can begeometrized on nonholonomic (equivalently, anholonomic) Riemannian manifolds modeling La-grange or Finsler spaces. In this context, the application of the Ricci flow theory and Perelman’ssuperb work on the functional approach to it may lead to novel geometric configurations and con-clusions of physical interest. Following his earlier work on this subject, the author sets out toinvestigate thermodynamic models related to certain classes of Lagrange and Finsler spaces.

After introducing anN -anholonomic manifold and itsN -connection, the main results on metriccompatible models of Lagrange and Finsler geometry on nonholonomic manifolds are reviewedleading to the important conclusion that any regular Lagrange mechanics (Finsler geometry) canbe modeled in two equivalent canonical forms as a nonholonomic Riemannian space or as anN -anholonomic Riemann-Cartan space with the metric and connection structures defined by thefundamental Lagrange (Finsler) function. Next, the author presents Perelman’s fundamental ideathat the Ricci flow not only is a gradient flow but also can be defined as a dynamical system onthe spaces of Riemannian metrics by introducing two functionals of Lyapunov type, and he showshow Perelman’s approach can be extended (by developing anN -adapted variational calculus) forN -anholonomic manifolds if the canonicald-connection (which is metric compatible with theLagrange canonical metric structure) is considered. With this equipment, theN -adapted evolutionequations for Lagrange-Ricci or Finsler-Ricci systems, i.e. for Lagrange and Finsler geometries,are derived. Finally, based on the analogy of Perelman’s functional to entropy, it is shown thatthis analogy holds also for nonholonomic Ricci flows. Consequently, a statistical/thermodynamicmodel for regular Lagrange mechanical systems is constructed whereby the expressions for theaverage energy, the entropy, and the energy fluctuation are explicitly given by use of the relevantpartition function. Actually, one could summarize these results as a new research programmeproposed for the geometrization of Lagrange mechanics following the theory of nonholonomicRicci flows and Lagrange-Finsler spaces equipped with compatible metric, nonlinear connection,and linear connection structures.

Reviewed byTheophanes Grammenos

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1. R. S. Hamilton:J. Diff. Geom.17 (1982), 255–306.MR0664497 (84a:53050)2. G. Perelman: The Entropy Formula for the Ricci Flow and its Geometric Applications,

math.DG/0211159.

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3. G. Perelman, Ricci flow with surgery on three-manifolds, math.DG/0303109.4. G. Perelman: Finite extinction time for the solutions to the Ricci flow on certain three-

manifolds, math.DG/0307245.5. H.-D. Cao and X.-P. Zhu:Asian J. Math.,10 (2006), 165–495.MR2233789 (2008d:53090)6. B. Kleiner and J. Lott: Notes on Perelman’s Papers, math.DG/0605667.7. J. W. Morgan and G. Tian: Ricci Flow and the Poincare Conjecture, math.DG/0607607.8. S. Vacaru, Nonholonomic Ricci Flows: I. Riemann Metrics and Lagrange-Finsler Geometry,

math.DG/06121629. S. Vacaru:Int. J. Mod. Phys.A21 (2006), 4899–4912.MR2265034 (2007g:53074)

10. R. Miron R. and M. Anastasiei:Vector Bundles and Lagrange Spaces with Applications toRelativity,(Geometry Balkan Press, Bukharest 1997); translation from Romanian (Ed. Acad.Romane, 1984).MR1469973 (98g:53002)

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13. Clifford and Riemann-Finsler Structures in Geometric Mechanics and Gravity, Selected Works,by S. Vacaru, P. Stavrinos, E. Gaburov and D. Gonta. Differential Geometry - DynamicalSystems, Monograph 7 (Geometry Balkan Press, 2006); www.mathem.pub.ro/dgds/mono/va-t.pdf and gr-qc/0508023.MR2255045 (2008i:53107)

14. S. Vacaru: Finsler and Lagrange geometries in Einstein and string gravity,Int. J. Geom. Meth-ods. Mod. Phys.5 (2008), 473–511.MR2428807 (2009f:53027)

15. D. Bao, S. S. Chern and Z. Chen:An Introduction to Riemann-Finsler Geometry,Springer,2000.MR1747675 (2001g:53130)

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17. S. Vacaru and M. Visinescu:Romanian Reports in Physics60 (2008), 218–238.18. G. Ruppeiner:Rev. Mod. Phys.67 (1995), 605–659;68 (1996), 313 (E).MR1349081

(96e:80004)19. R. Mrugala, J. D. Nulton, J. C. Schon, and P. Salamon:Phys. Rev.A 41 (1990), 3156–3160.

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with Applications in Physics and Biology,Kluwer, 1993.MR1273129 (95e:53094)22. D. L. Rapoport:Rep. Math. Phys.50 (2002), 211–250.MR1936743 (2003i:58070)23. S. Vacaru,Ann. Phys.(N. Y.) 290(2001), 83–123.MR1834616 (2002d:82064)24. C. Tsallis, A. Rapisarda, A. Pluchino, E. P. Borges:PhysicaA 381 (2007), 143–147.25. S. Vacaru:J. Math. Phys.49 (2008), 043504.MR2412299 (2009d:53091)26. S. Vacaru: Nonholonomic Ricci Flows,Exact Solutions in Gravity, and Symmetric and Non-

symmetric Metrics,to be published in:Int. J. Theor. Phys.47(2008), arXiv: 0806.3812 [gr-qc].MR2485601 (2010b:53124)

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27. S. Vacaru: Spectral Functionals, Nonholonomic Dirac Operators, and Noncommutative RicciFlows, arXiv: 0806.3814 [math-ph].MR2548633

Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

c© Copyright American Mathematical Society 2010, 2011

Article

Citations

From References: 2From Reviews: 0

MR2485601 (2010b:53124)53C44 (53C80 83C15)

Vacaru, Sergiu I. (3-FLDS2)Nonholonomic Ricci flows, exact solutions in gravity, and symmetric and nonsymmetricmetrics. (English summary)Internat. J. Theoret. Phys.48 (2009),no. 2,579–606.

The aim of the present work, belonging to an excellent series of papers by the author on non-holonomic Ricci flows, is to provide a geometric motivation for gravity models with nonsym-metric metrics. The geometric methods presented have already been elaborated in generalizedLagrange and Finsler geometries. However, the present work is restricted to (pseudo-)Riemannianand Riemann-Cartan spaces with effective torsion induced by nonholonomic frames, and to non-symmetric metrics induced by Ricci flows. The subject becomes increasingly interesting as thereis no physical principle whatsoever prohibiting the consideration of theories with nonsymmetricmetrics. The author starts with a geometric formulation for the systems of evolution equationswith non-holonomic Ricci flows transforming symmetric metrics into nonsymmetric ones. So,after presenting the notion of non-holonomic manifolds equipped with a nonlinear connectionstructure (i.e., N-anholonomic manifolds), the non-holonomic Ricci flow equations are given withrespect to N-adapted frames. Using the anholonomic frame method and introducing a specificoff-diagonal metric ansatz, exact non-holonomic Ricci flow solutions are constructed for grav-ity with symmetric and nonsymmetric metric components in the case of non-holonomic Einsteinspaces. Next, the anholonomic frame method is applied in order to construct Ricci flow solutionsdescribing how nonholonomic deformations of 4D Taub-NUT spaces may lead to nonsymmetricmetric configurations if the flow parameter is associated to a time-like coordinate for pp-waves.The anholonomic frame method is further applied to generate 4D vacuum metrics with nontriv-ial antisymmetric terms defined by nonlinear pp-waves and solitonic interactions for vanishingsources and the Levi-Civita connection. It is also shown that Ricci flows subjected to correspond-ing nonholonomic deformations of the Schwarzschild metric lead to nonsymmetric metrics. Itshould be pointed out that the results obtained provide a strong support for the significance andusefulness of the non-holonomic Ricci flow evolution for gravitational systems. Indeed, it pro-vides an impressive geometric ground for studying gravity theories with nonsymmetric metric

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structure.Reviewed byTheophanes Grammenos

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MR2492620 (2010f:53163)53D55 (53C60 81S10)

Anastasiei, Mihai (R-IASIM) ; Vacaru, Sergiu I. (3-FLDS2)Fedosov quantization of Lagrange-Finsler and Hamilton-Cartan spaces and Einstein gravitylifts on (co) tangent bundles. (English summary)J. Math. Phys.50 (2009),no. 1,013510, 23pp.

Summary: “We provide a method of converting Lagrange and Finsler spaces and their Legendretransforms to Hamilton and Cartan spaces into almost Kahler structures on tangent and cotangentbundles. In particular cases, the Hamilton spaces contain nonholonomic lifts of (pseudo) Rie-mannian/Einstein metrics on effective phase spaces. This allows us to define the correspondingFedosov operators and develop deformation quantization schemes for nonlinear mechanical andgravity models on Lagrange- and Hamilton-Fedosov manifolds.”

In this context the results of [A. V. Karabegov and M. Schlichenmaier, Lett. Math. Phys.57(2001), no. 2, 135–148;MR1856906 (2002f:53156)] are modified for almost Kahler manifoldsendowed with the above-mentioned additional canonical geometric structures.

Reviewed byMartin Schlichenmaier

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Article

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MR2479265 (2010b:37194)37K25 (35Q53 37K05 37K10 53C60)

Anco, Stephen C.(3-BRCK); Vacaru, Sergiu I. (R-IASIM)Curve flows in Lagrange-Finsler geometry, bi-Hamiltonian structures and solitons. (Englishsummary)J. Geom. Phys.59 (2009),no. 1,79–103.

Summary: “Methods in Riemann-Finsler geometry are applied to investigate bi-Hamiltonian struc-tures and related mKdV hierarchies of soliton equations derived geometrically from regular La-grangians and flows of non-stretching curves in tangent bundles. The total space geometry andnonholonomic flows of curves are defined by Lagrangian semisprays inducing canonical nonlinearconnections (N -connections), Sasaki-type metrics and linear connections. The simplest examplesof such geometries are given by tangent bundles on Riemannian symmetric spacesG/SO(n) pro-vided with anN -connection structure and an adapted metric, for which we elaborate a completeclassification, and by generalized Lagrange spaces with constant Hessian. In this approach, bi-Hamiltonian structures are derived for geometric mechanical models and (pseudo) Riemannianmetrics in gravity. The results yield horizontal and vertical pairs of vector sine-Gordon equationsand vector mKdV equations, with the corresponding geometric curve flows in the hierarchies de-scribed in an explicit form by nonholonomic wave maps and mKdV analogs of nonholonomicSchrodinger maps on a tangent bundle.”

Reviewed byArtur Sergyeyev

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14. S.C. Anco, bi-Hamiltonian operators, integrable flows of curves using moving frames, andgeometric map equations, J. Phys. A. 39 (2006) 2043–2072.MR2211976 (2007g:37051)

15. C. Athorne, Some integrable equations associated with symmetric spaces, in: Nonlinear Evolu-tions [Balaruc-les-Bains, 198], World Sci. Publ. Teaneck, NJ, 1988, pp. 191–198.MR1031507(91d:58088)

16. C. Athorne, Local Hamiltonian structures of multicomponentKdV equations, J. Phys. A. 21(1988) 4549–4556.MR0974701 (90b:58061)

17. S.C. Anco, Hamiltonian flows of curves inG/SO(n) and vector soliton equations ofmKdVand sine-Gordon Type, Symmetry, Integrability and Geometry: Methods and Applications 2(2006) 044.MR2217753 (2007a:37073)

18. J.-P. Wang, Generalized Hasimoto Transformation and Vector Sine–Gordon Equation, in: S.Abenda, G. Gaeta, S. Walcher (Eds.), SPT 2002: Symmetry and Perturbation Theory (CalaGonone), WorldScientific, River Edge, 2002, pp. 276–283.MR1976678 (2004c:37172)

19. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Providence, Amer.Math. Soc, 2001.MR1834454 (2002b:53081)

20. V.V. Sokolov, T. Wolf, Classification of integrable vector polynomial evolution equation, J.Phys. A. 34 (2001) 11139–11148.MR1872986 (2002j:35275)

21. S.C. Anco, T. Wolf, Some symmetry classifications of hyperbolic vector evolution equations, J.Nonlinear Math. Phys. 12 (suppl. 1) (2005) 13–31; J. Nonlinear Math. Phys. 12 (2005) 607–608(erratum).MR2117166 (2005i:35171)

22. S. Vacaru, Curve flows and solitonic hierarchies generated by (semi) Riemannian metrics..23. A. Bejancu, Finsler Geometry and Applications, Ellis Horwood, Chichester, England, 1990.

MR1071171 (91i:53075)24. A. Bejancu, H.R. Farran, Foliations and Geometric Structures, Springer, 2005.MR2190039

(2006j:53034)25. S. Vacaru, Exact solutions with noncommutative symmetries in Einstein and gauge Gravity, J.

Math. Phys. 46 (2005) 042503.MR2131241 (2006b:83128)26. S. Vacaru, P. Stavrinos, E. Gaburov, D. Gonta, Clifford and Riemann–Finsler Structures

in Geometric Mechanics and Gravity. Selected Works, in: Differential Geometry–Dynamical Systems, Monograph, vol. 7, Geometry Balkan Press, Bucharest, 2006www.mathem.pub.ro/dgds/mono/vat.pdfgr-qc/0508023.MR2255045 (2008i:53107)

27. K. Yano, S. Ishihara, Tangent and Cotangent Bundles, M. Dekker, Inc., New York, 1978.28. S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vols. I and II, Wiley, 1969.

MR1393940 (97c:53001a)29. R.W. Sharpe, Differential Geometry, Springer–Verlag, New York, 1997.MR1453120

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(98m:53033)30. G. Vranceanu, Sur les espaces non holonomes, C. R. Acad. Paris 103 (1926) 852–854.31. G. Vranceanu, Lecons de Geometrie Differentielle, Vol II (Edition de l’Academie de la Re-

publique Populaire de Roumanie, 1957).MR0124823 (23 #A2133)32. D. Bao, S.-S. Chern, Z. Shen, An Introduction to Riemann–Finsler Geometry, in: Graduate

Texts in Math, vol. 200, Springer–Verlag, 2000.MR1747675 (2001g:53130)33. S.C. Anco, Hamiltonian curve flows in Lie groupsG ⊂ U(N) and vector NLS,mKdV,

sine-Gordon soliton equations, in: IMA Volumes in Mathematics and its Applications, in:Symmetries and Overdetermined Systems of Partial Differential Equations, vol. 144, AMS,2007, pp. 223–250.MR2384712 (2008m:37122)

34. S.C. Anco, Group-invariant soliton equations and bi-Hamiltonian geometric curve flows inRiemannian symmetric spaces, J. Geom. Phys. 58 (2008) 1–37.MR2378453 (2010a:37128)

35. A. Sergyeyev, Why nonlocal recursion operators produce local symmetries: New results andapplications, J. Phys. A. 38 (2005) 3397–3407.MR2132718 (2005m:37166)

36. C. Barcelo, S. Liberati, M. Visser, Analogue gravity, Living Rev. Rel. 8 (2005) 12.37. R. Miron, D. Hrimiuc, S. Shimada, V.S. Sabau, The Geometry of Hamilton and Lagrange

Spaces, Kluwer Academic Publishers, Dordrecht, Boston, London, 2000.38. E. Cartan, Les Espaces de Finsler, Hermann, Paris, 1935.39. M. Matsumoto, Foundations of Finsler Geometry and Special Finsler Spaces, Kaisisha, Shi-

gaken, 1986.MR0858830 (88f:53111)40. S. Vacaru, Interactions, Strings and Isotopies in Higher Order Anisotropic Superspaces,

Hadronic Press, Palm Harbor, FL, USA, 1998.MR1679228 (2000c:81343)41. F. Etayo, R. Santamaria, S. Vacaru, Lagrange–Fedosov nonholonomic manifolds, J. Math.

Phys. 46 (2005) 032901.MR2125569 (2006a:53025)42. S. Vacaru, Curve flows and solitonic hierarchies generated by Einstein metrics. arXiv:

0810.0707 [math-ph].MR260164443. S. Vacaru, Ricci flows and solitonicpp–waves, Internat. J. Modern Phys. A 21 (2006) 4899–

4912.MR2265034 (2007g:53074)44. S. Vacaru, Nonholonomic Ricci flows: II. Evolution equations and dynamics, J. Math. Phys. 49

(2008) 043504.MR2412299 (2009d:53091)45. S. Vacaru, Nonholonomic Ricci flows: III. Curve flows and solitonic hierarchies. arXiv:

0704.2062 [math.DG].46. S. Vacaru, Deformation quantization of almost Kahler models and Lagrange-Finsler spaces, J.

Math. Phys. 48 (2007) 123509.MR2377841 (2008k:53209)47. S. Vacaru, Deformation quantization of nonholonomic almost Kahler models and Einstein

gravity, Phys. Lett. A 372 (2008) 2949–2955.MR2404752 (2009g:53137)48. S. Vacaru, Finsler and Lagrange geometries in Einstein and string gravity, Int. J. Geom. Methods

Mod. Phys. (IJGMMP) 5 (2008) 473–511.MR2428807 (2009f:53027)Note: This list reflects references listed in the original paper as accurately as possible with no

attempt to correct errors.

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Article

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MR2483701 (2010i:53039)53C08 (53C80 58J20 81T13)

Vacaru, Sergiu I. (3-FLDS2); Gonzalez-Hernandez, Juan F.(R-IASIM)Nonlinear connections on gerbes, Clifford-Finsler modules and the index theorems. (Englishsummary)Indian J. Math.50 (2008),no. 3,573–606.

Summary: “The geometry of nonholonomic bundle gerbes, provided with nonlinear connectionstructure, and nonholonomic gerbe modules is elaborated as the theory of Clifford modules onnonholonomic manifolds which fail to be Spin. We explore an approach to such nonholonomicDirac operators and derive the related Atiyah-Singer index formulæ. There are considered certainapplications in modern gravity and geometric mechanics of such Clifford-Lagrange/Finsler gerbesand their realizations as nonholonomic Clifford and Riemann-Cartan modules.”

Reviewed byJose A. Vallejo(San Luis Potosı)

c© Copyright American Mathematical Society 2010, 2011

Article

Citations

From References: 1From Reviews: 0

MR2470525 (2009i:83102)83D05 (53C60)

Vacaru, Sergiu I. (3-FLDS2)Einstein gravity, Lagrange-Finsler geometry, and nonsymmetric metrics. (Englishsummary)SIGMA Symmetry Integrability Geom. Methods Appl.4 (2008),Paper071, 29pp.

This paper continues a series of studies that the author has devoted to nonlinear connections(defined by nonintegrable distributions) and geometric constructions adapted to them (such asLagrange-Finsler, Clifford and Lie algebroid structures, metrics, etc.). This time, given a nonlin-ear connectionN and a nonsymmetric metricg, a detailed study of properties such as torsionand curvature, compatibility between metrics and connections, etc., is made, mimicking the well-known theorems of the Riemannian case. The motivation to consider nonsymmetric tensorsgcomes from physics (see [J. W. Moffat, Phys. Lett. B355(1995), no. 3-4, 447–452;MR1345596(96h:83039)], where an old idea of Einstein to build a unified setting for gravitation and electro-magnetism was developed). In the last sections of the paper, the author explores the relationshipwith gravitation, recasting the metric-noncompatible formulation of Moffat in a metric-compatibleform (with the nonsymmetricg). He also explores the connection with other branches of mathe-

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matics (Lagrange-Finsler spaces and Ricci flows).Reviewed byJose A. Vallejo(San Luis Potosı)

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Article

Citations

From References: 6From Reviews: 0

MR2428807 (2009f:53027)53B40 (53B50 53C80 83C99 83D05)

Vacaru, Sergiu I. (3-FLDS2)Finsler and Lagrange geometries in Einstein and string gravity. (English summary)Int. J. Geom. Methods Mod. Phys.5 (2008),no. 4,473–511.

This is a survey on Finsler-Lagrange geometry and the anholonomic method in general relativityand gravitation. The geometrical structures presented in the paper refer to the following top-ics: nonholonomic manifolds, Finsler-Lagrange spaces, nonlinear connections, Riemann-Cartanspaces, and Hermitian manifolds. By using this large variety of important topics from differentialgeometry, the author succeeds in presenting new equivalent (non)holonomic formulations of grav-ity theories and in constructing exact solutions, with effective Lagrange and Finsler structures, inEinstein and string gravity. Also, he emphasizes that the Einstein gravity can be equivalently refor-mulated in terms of almost Hermitian geometry with preferred frame structure. Finally, the authorpresents explicit examples of exact solutions in Einstein and string gravity. The examples givenfor string gravity show the local anisotropic configurations describing gravitational solitonic pp-waves and their effective Lagrange spaces. Most of the results presented in the paper have beenobtained by the author and his collaborators. The paper brings both light and hope to the difficulttask of applying generalizations of (pseudo)Riemannian geometry to physics.

Reviewed byAurel Bejancu

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Article

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MR2412299 (2009d:53091)53C44 (37J60 53C60)

Vacaru, Sergiu I. (3-FLDS2)Nonholonomic Ricci flows. II. Evolution equations and dynamics. (English summary)J. Math. Phys.49 (2008),no. 4,043504, 27pp.

In a series of five successive, densely written papers under the general title “Nonholonomic Ricciflows” (part V of the series appeared in May 2007 as a preprint [arxiv.org/abs/0705.0729]), theauthor has studied the theory of nonholonomic Ricci flows, i.e., Ricci flows for non-Riemannianmanifolds, as well as relevant applications of the theory in classical and quantum physics andgeometric mechanics. The present paper, which is the second of the series, aims at investigatingthe evolution of geometrical objects under Ricci flows with nonholonomic constraints on the framestructure.

The present study starts with the formulation of the Hamilton-Perelman theory of Ricci flowas an evolution equation onN -nonholonomic manifolds, i.e., nonholonomic manifolds equippedwith a nonlinear connection structure. Although, in general, symmetric metrics may evolve intononsymmetric metrics under nonholonomic Ricci flows, for reasons of simplicity only those non-holonomic evolutions are examined that result in symmetric metrics. The evolution equations arestudied for two equivalent metric compatible (defined by the same metric structure) linear connec-tions: the Levi-Civita connection and the canonical distinguished connection (a linear connectionthat preserves under parallelism the Whitney sum and contains nontrivial torsion coefficients). Thecentral idea behind the aforesaid study is that if a geometric Ricci flow construction is well definedfor one of the two connections, it can be equivalently redefined for the other connection by con-sidering the so-called distortion tensor. As examples ofN -nonholonomic Ricci flows, the authorexamines nonholonomic Einstein spaces andN -nonholonomic Ricci solitons. Furthermore, theexistence and uniqueness of theN -nonholonomic evolution is proved. Perelman’s functionals aredefined onN -nonholonomic manifolds for the canonical distinguished connection. AnN -adaptedvariational calculus is constructed and applied to Perelman’s functionals. In this way, a rigorousproof of the evolution equations for generalized Finsler-Lagrange and nonholonomic metrics isobtained, while properties of the associated distinguished energy for nonholonomic configurationsare examined and certain rules for the extension of the proofs for the Levi-Civita connections tothe canonical distinguished connections are given. Further, a statistical analogy for nonholonomicRicci flows is developed, in the context of which properties of theN -nonholonomic entropy areexamined by use of Perelman’sW -functional, and related thermodynamic expressions for the av-erage energy, the entropy, and the fluctuation are obtained by computing the partition function forthe canonical ensemble for compact configurations.

Finally, two applications in physics are discussed. First, the nonholonomic Ricci flow evolutionfor solitonic pp-wave solutions of vacuum Einstein field equations in general relativity is studied. Infact, in this context a new class of four-dimensional solutions (metrics) is constructed by using thenonholonomic frame approach. The second application consists in the computation of Perelman’sentropy for classes of regular Lagrange systems (mechanical, or analogous gravitational). It isto be pointed out, however, that the physical interpretation of the thermodynamical expressionsobtained by using Perelman’s functionals remains an open question.{For Part I see [S. I. Vacaru, “Nonholonomic Ricci flows. I. Riemann metrics and Lagrange-

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Finsler geometry”, preprint, arxiv.org/abs/math/0612162].}Reviewed byTheophanes Grammenos

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MR1375255 (97e:53075)3. G. Perelman, e-print arXiv:math.DG/0211159.MR2334184 (2008k:53141)4. G. Perelman, e-print arXiv:math.DG/0309021.MR2334184 (2008k:53141)5. G. Perelman, e-print arXiv:math.DG/0307245.MR2334184 (2008k:53141)6. H.-D. Cao and X.-P. Zhu, Asian J. Math.10, 165 (2006), e-print arXiv:math.DG/0612069.

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(International Press, Somerville, 2003).MR2145154 (2006e:53002)8. B. Kleiner and J. Lott, e-print arXiv:math.DG/0605667.9. J. W. Morgan and G. Tian, e-print arXiv:math.DG/0607607.

10. S. Vacaru, e-print arXiv:math.DG/0612162.11. S. Vacaru, Int. J. Mod. Phys. A21,4899 (2006).12. S. Vacaru and M. Visinescu, Int. J. Mod. Phys. A22,1135 (2007).MR2311055 (2008c:53069)13. S. Vacaru and M. Visinescu, Rom. Rep. Phys.60,2 (2008), e-print arXiv:gr-qc/0609086.14. S. Vacaru, e-print arXiv:math.DG/0701621.15. S. Vacaru, J. Math. Phys.46,042503 (2005).MR2131241 (2006b:83128)16. Clifford and Riemann-Finsler Structures in Geometric Mechanics and Gravity, Selected Works,

edited by S. Vacaru, P. Stavrinos, E. Gaburov, and D. Gonta (Geometry Balkan, Bukharest,2006).MR2255045 (2008i:53107)

17. S. Vacaru, Int. J. Geom. Methods Mod. Phys. (to be published).18. S. Vacaru, J. Math. Phys.47,093504 (2006).MR2263658 (2007m:53108)19. S. Vacaru, Int. J. Geom. Methods Mod. Phys.4, 1285 (2007).MR2378384 (2008k:83030)20. R. Hamilton, Contemp. Math.71,237 (1988).MR0954419 (89i:53029)21. S. Vacaru, e-print arXiv:math.DG/0704.2062.22. D. De Turck, J. Diff. Geom.18,157 (1983).23. W. X. Shi, J. Diff. Geom.30,223 (1989).MR1001277 (90i:58202)24. O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Ural↩tseva,Linear and Quasilinear Equations

of Parabolic Type,Translation of Mathematical Monographs Vol. 23 (American MathematicalSociety, Providence, RI, 1968), Chap. XL, p. 648.MR0241822 (39 #3159b)

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26. A. Peres, Phys. Rev. Lett.3, 571 (1959).27. S. Vacaru and D. Singleton, Class. Quantum Grav.19,3583 (2002).MR1922140 (2003k:83122)28. V. A. Belinski and V. E. Zakharov, Sov. Phys. JETP48, 985 (1978) [Zh. Eksp. Teor. Fiz.75,

1955 (1978)].29. V. Belinski and E. Verdaguer,Gravitational Solitons(Cambridge University Press, Cambridge,

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2001).MR1839019 (2002d:83001)30. S. Vacaru, J. High Energy Phys. 04, 009 (2001).31. S. Vacaru and F. C. Popa, Class. Quantum Grav.18,4921 (2001).MR1869374 (2003b:83123)32. S. Vacaru, e-print arXiv:math-ph/0705.0728.33. S. Vacaru, e-print arXiv:math-ph/0705.0729.34. R. Miron and M. Anastasiei,The Geometry of Lagrange Spaces: Theory and Applications

(Kluwer Academic, Dordrecht, 1994), Vol. 59.MR1281613 (95f:53120)35. S. Vacaru, J. Math. Phys.48,123509 (2007).MR2377841 (2008k:53209)36. S. Vacaru, e-print arXiv:gr-qc/0801.4942.37. M. Anastasiei and S. Vacaru, e-print arXiv:math-ph/0710.3079.38. G. Ruppeiner, Rev. Mod. Phys.67,605 (1995);68,0313(E) (1995).MR1349081 (96e:80004)39. R. Mrugala, J. D. Nulton, J. C. Schon, and P. Salamon, Phys. Rev. A41, 3156 (1990).

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with Applications in Physics and Biology(Kluwer, Dordrecht, 1993).42. D. L. Rapoport, Rep. Math. Phys.50,211 (2002).MR1936743 (2003i:58070)43. S. Vacaru, Ann. Phys. (N.Y.)290,83 (2001).MR1834616 (2002d:82064)44. C. Castro, Adv. Studies Theor. Phys.1, 119 (2007).MR2346723 (2008f:83024)

Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

c© Copyright American Mathematical Society 2009, 2011

Article

Citations

From References: 6From Reviews: 2

MR2404752 (2009g:53137)53D55 (53B40 83C45)

Vacaru, Sergiu I. (3-FLDS2)Deformation quantization of nonholonomic almost Kahler models and Einstein gravity.(English summary)Phys. Lett. A372(2008),no. 17,2949–2955.

In the present paper the author demonstrates how the structure of a non-integrable distribution onan even-dimensional manifoldV 2n together with a (pseudo-)Riemannian structure allows one toendowV 2n with the structure of an almost Kahler manifold. Furthermore, he constructs a certainlinear connection inTV 2n which is compatible with the above (pseudo-)Riemannian metric but ingeneral has non-vanishing torsion components.

After these steps the author is in the position to apply the Fedosov construction [cf. B. V. Fedosov,J. Differential Geom.40 (1994), no. 2, 213–238;MR1293654 (95h:58062)] for deformationquantizations on almost Kahler manifolds given in [A. V. Karabegov and M. Schlichenmaier, Lett.

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Math. Phys.57 (2001), no. 2, 135–148;MR1856906 (2002f:53156)].Hence, the main aim of the paper under review is to show that an even-dimensional spacetime, in

addition to a non-integrable distribution, naturally gives rise to a deformation quantization of thatspacetime. Nevertheless, this should not be confused with an honest quantization of the underlyinggravitational field.

Finally, there is a misprint in the recursion formula forr(k+3) on page 2953, where one shouldread

− i

v

k∑l=0

r(k−l+2) ◦ r(l+2)

rather than

− i

v

k∑l=0

r(l+2) ◦ r(l+2).

Reviewed byNikolai A. Neumaier

c© Copyright American Mathematical Society 2009, 2011

Article

Citations

From References: 1From Reviews: 0

MR2522405 (2010i:53178)53D55 (53C60 81S10 83C45)

Vacaru, Sergiu I. (3-FLDS2)Generalized Lagrange transforms: Finsler geometry methods and deformation quantizationof gravity. (English summary)An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.)53 (2007),suppl. 1,327–342.

The author aims to generalize some results from a previous paper [S. I. Vacaru, J. Math. Phys.48(2007), no. 12, 123509, 14 pp.;MR2377841 (2008k:53209)] by viewing an almost Kahler manifoldas a generalized Lagrange space. Most of the work done here is contained in the aforementionedpaper and another one by the author [Phys. Lett. A372(2008), no. 17, 2949–2955;MR2404752(2009g:53137)]. The paper seems to be based on the notion of a fibered manifold (as nearlyeverything is written in fibered coordinates) of dimensionn + n, although the author claims thathis results are valid for a general2n-dimensional manifoldV 2n.

Reviewed byJose A. Vallejo(San Luis Potosı)

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MR2378384 (2008k:83030)83C15Vacaru, Sergiu I. (3-FLDS2)Parametric nonholonomic frame transforms and exact solutions in gravity. (Englishsummary)Int. J. Geom. Methods Mod. Phys.4 (2007),no. 8,1285–1334.

The author has, in many previous publications, together with various collaborators, developed amethod of nonholonomic frame transformations in (pseudo-)Riemannian manifolds with the aimof generating new solutions of Einstein’s equations from an explicitly known one. The techniqueis inspired by the papers of R. Geroch [J. Mathematical Phys.12 (1971), 918–924;MR0286442(44 #3651); J. Mathematical Phys.13 (1972), 394–404;MR0300598 (45 #9643)].

The paper under review summarizes the formalism of nonholonomic frame transformations,i.e., transformations of the basis of each tangent space which is assumed split into a horizontalsubspace and its vertical complement such that under these transformations the vertical spaceremains invariant. The idea is to start with a metric admitting one or more local isometries and ofa simple form in some coordinates and to replace the differentials of some ‘vertical’ coordinatesby co-frames that include some of the differentials of ‘horizontal’ coordinates (thus changing theN-connection). When the coefficients depend only on one or a few of the coordinates and possiblysome other parameters and the original metric satisfies Einstein’s or some other field equationsthen the deformed metric will satisfy the field equations provided some hopefully much simplerpartial differential equations hold for the few introduced coefficients of the frame transformations.A number of examples are given in four and five dimensions starting with the Schwarzschild andthe pp-wave exact solutions of the vacuum Einstein equations.

Reviewed byHans-Peter Kunzle

c© Copyright American Mathematical Society 2008, 2011

Article

Citations

From References: 5From Reviews: 2

MR2377841 (2008k:53209)53D55 (53C60)

Vacaru, Sergiu I. (3-FLDS2)Deformation quantization of almost Kahler models and Lagrange-Finsler spaces. (Englishsummary)J. Math. Phys.48 (2007),no. 12,123509, 14pp.

In a very interesting paper [Lett. Math. Phys.57 (2001), no. 2, 135–148;MR1856906(2002f:53156)] A. V. Karabegov and M. Schlichenmaier extended Fedosov’s techniques of quan-

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tization of symplectic manifolds to the case of almost Kahler manifolds. By using the fact that aFinsler space can be modeled as an almost Kahler manifold with some specific features, the au-thor of the paper under review is able to rewrite the results of Karabegov and Schlichenmaier,expressing them in the language of Finsler geometry.

Reviewed byJose A. Vallejo(San Luis Potosı)

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(Kluwer Academic, Dordrecht, 1994), FTPH No. 59.MR1281613 (95f:53120)8. R. Miron and Gh. Atanasiu, Seminarul de Mecanica (unpublished), Vol. 40, p. 1; Revue

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22. F. Etayo, R. Santamarıa, and S. Vacaru, J. Math. Phys.46, 032901 (2005).MR2125569(2006a:53025)

23. S. Vacaru, e-print arXiv:0707.1526.24. S. Vacaru, e-print arXiv:0707.1667.25. Proofs consist of straightforward computations.26. For simplicity, in this work, we shall omit left labelsL in formulas if it will not result in

ambiguities; we shall use boldface indices for spaces and objects provided or adapted to anN -connection structure.

27. We contract horizontal and vertical indices following the rulei = 1 is a = n + 1; i = 2 is a =n +2;...; i = n is a = n +n′′.

28. E. Cartan,Les Espaces de Finsler(Hermann, Paris, 1935).29. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternhemer, Lett. Math. Phys.1,521

(1977).30. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternhemer, Ann. Phys. (N.Y.)111,

61 (1978).31. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternhemer, Ann. Phys. (N.Y.)111,

111 (1978).32. P. Deligne and P. Lecompte, Selecta Math., New Ser.1, 667 (1995).MR1383583 (97a:58068)33. R. Nest and B. Tsygan, Adv. Math.11,223 (1995).MR1337107 (96j:58163a)34. F. Berezin, Math. USSR, Izv.8, 1109 (1974).35. M. Bordermann, E. Meinrenken, and M. Schlichenmaier, Commun. Math. Phys.165, 281

(1994).MR1301849 (96f:58067)36. V. Dolgushev, S. Lyakhovich, and A. Sharapov, Nucl. Phys. B606,647 (2001).MR1847979

(2002d:53124)37. A. Karabegov, Commun. Math. Phys.180,745 (1996).MR1408526 (97k:58072)38. A. Karabegov, Lett. Math. Phys.43,347 (1998).MR1620745 (99f:58086)39. M. Kontsevich, Lett. Math. Phys.66,157 (2003).MR2062626 (2005i:53122)40. M. Kontsevich, Lett. Math. Phys.48,35 (1999).MR1718044 (2000j:53119)41. K. Yano,Differential Geometry on Complex and Almost Complex Spaces(Pergamon, New

York, 1965).MR0187181 (32 #4635)42. We chose a definition of this tensor as in Ref. 6 which is with minus sign compared to the

definition used in Ref. 21; we also use a different letter for this tensor, like for theN -connectioncurvature, because in our case, the symbolN is used forN connections.

43. This formula is a nonholonomic analog, for our conventions, with inverse sign, of formula (2.9)from Ref. 21.

44. S. Vacaru, J. Math. Phys.46,042503 (2005).MR2131241 (2006b:83128)45. S. Vacaru, J. Math. Phys.47,093504 (2006).MR2263658 (2007m:53108)46. S. Vacaru, Nucl. Phys. B434,590 (1997).MR1453774 (98j:81303)47. S. Vacaru, J. High Energy Phys.09,011 (1998).48. It should be noted that formulas (20)—(24) can be written for any metricg and metric com-

patibled connectionD, Dg = 0, onTM , provided with arbitraryN connectionN (we haveto omit hats and labelsL. It is a more sophisticated problem to define such constructions for

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Finsler geometries with the so-called Chern connection which are metric noncompatible (Ref.51). For applications in standard models of physics, we chose the variants of Lagrange-Finslerspaces defined by metric compatibled connections, see discussion in Ref. 13.

49. M. Bordemann and St. Waldmann, Lett. Math. Phys.41,243 (1997).MR1463874 (98h:58069)50. S. Vacaru, P. Stavrinos, E. Gaburov, and D. Gonta,Clifford and Riemann-Finsler Structures in

Geometric Mechanics and Gravity,Selected Works, Differential Geometry—Dynamical Sys-tems Monograph 7 (Geometry Balkan Press, Bucharest, 2006) (www.mathem.pub.ro/dgds/mono/va-t.pdf and gr-qc/0508023).MR2255045 (2008i:53107)

51. D. Bao, S.-S. Chern, and Z. Shen,An Introduction to Riemann-Finsler Geometry,GraduateTexts in Mathematics Vol. 200 (Springer-Verlag, Berlin, 2000).MR1747675 (2001g:53130)

Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

c© Copyright American Mathematical Society 2008, 2011

Article

Citations

From References: 3From Reviews: 0

MR2311055 (2008c:53069)53C44 (83C15)

Vacaru, Sergiu I. (3-FLDS2); Visinescu, MihaiNonholonomic Ricci flows and running cosmological constant. I. 4D Taub-NUT metrics.(English summary)Internat. J. Modern Phys. A22 (2007),no. 6,1135–1159.

In this paper, the formalism of the “anholonomic frame method” is elaborated and applied to theconstruction of exact solutions of Ricci flow equations describing nonholonomic deformationsof off-diagonal metrics and Taub-NUT-like metrics. After presenting the general form of genericoff-diagonal metrics as usually considered in anholonomic manifolds with associated nonlinearconnection structure, a general class of the corresponding anholonomic Ricci flow equations isformulated and a brief outline of the method to construct their solutions is given. The generation ofsuch solutions is simpler for Levi-Civita connections and it is presented as a study of the emergingconstraints in this case. The anholonomic frame formalism is then applied to the generation andanalysis of exact solutions describing off-diagonal Ricci flows of four-dimensional Taub-NUT-likemetrics. In particular, topological Taub-NUT-AdS/dS spacetimes are considered, which describephysical settings including a nontrivial cosmological constant. A very useful appendix contains anoutline of the geometry of nonlinear connections (Riemann-Cartan manifolds) and anholonomicdeformations.

As a conclusion, it can be said that the construction of exact off-diagonal solutions that dependboth holonomically and anholonomically on two and three variables is a mathematically difficultand tedious task, but the anholonomic frame method provides a possibility of treating this problemdespite the circumstantially obscure physical and/or conceptual interpretations of the results one

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obtains in this way.Reviewed byTheophanes Grammenos

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MR2265034 (2007g:53074)53C44 (53C80 83C15)

Vacaru, Sergiu I.Ricci flows and solitonic pp-waves. (English summary)Internat. J. Modern Phys. A21 (2006),no. 23-24,4899–4912.

Summary: “We find exact solutions describing Ricci flows of four-dimensional pp-waves non-linearly deformed by two-/three-dimensional solitons. Such solutions are parametrized by five-dimensional metrics with generic off-diagonal terms and connections with nontrivial torsion whichcan be related, for instance, to antisymmetric tensor sources in string gravity. Nontrivial limits tofour-dimensional configurations and Einstein gravity are defined.”

Reviewed byWeimin Sheng

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Article

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MR2263658 (2007m:53108)53D17 (53B15 58H99)

Vacaru, Sergiu I. (3-BRCK)Clifford-Finsler algebroids and nonholonomic Einstein-Dirac structures. (Englishsummary)J. Math. Phys.47 (2006),no. 9,093504, 20pp.

A real Lie algebroid(E, [·, ·]E, ρE) consists of a smooth real vector bundle over a manifoldM ,τ :E →M , endowed with a real Lie algebra structure[·, ·]E on the vector spaceΓ(E) of smoothglobal sections ofE, and a vector bundle morphismρE:E → TM (called the anchor map) suchthat(1) [X, fY ]E = f [X, Y ]E +(ρE(X)f) ·Y for all X, Y ∈ Γ(E) andf ∈ C∞(M),(2) ρE defines a Lie algebra morphism from the Lie algebra(Γ(E), [·, ·]E) into (X(M), [·, ·]).Complex Lie algebroids can be defined similarly, with the tangent bundleTM replaced by the

complexified tangent bundleTCM .

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In this paper, the author studies the consequences of having some additional structures onMandE. First, it is assumed that on the tangent bundlep:TM → M there is defined a non-linearconnection, understood as a smooth splitting of the exact sequence

0→ vTM ↪→ TTMp′−→ p−1TM → 0

of vector bundles over the manifoldTM (herep−1TM is the pullback ofTM , p′ is defined byfactoring the tangent mapp∗:TTM → TM throughp−1TM and vTM = ker p′). It must beremarked that the notation in the paper is slightly different (and less standard). The author callsa manifoldM with such a non-linear connection onTM an anholonomic manifold (to stress thefact that, in general, the non-linear connection is non-integrable, or non-holonomic).

Second, the author takesE to be the Clifford bundle ofM , E = Cl(TM) (thus presupposingthe existence of a pseudo-Riemannian metricg). The anchor map, denotedsρ, is now a morphismsρ: Cl(TM) → TM , and the author defines it in a way that makes it “compatible” with thenon-linear connection onTM . The resulting Lie algebroid is called a Clifford algebroid.

After discussing the construction of several differential-geometric notions on manifolds with anon-linear connection, such as covariant differentiation, Dirac operators, etc., the author analyzessome field equations of relevance in physics: scalar, Proca, graviton, spinor and gauge fieldinteractions.

However, some comments about somewhat inaccurate statements in the paper are in order. Withrespect to the title: although the paper is called “Clifford-Finsler algebroids and nonholonomicEinstein-Dirac structures”, the prospective reader should know that:(1) Finsler structures do not appear anywhere, except for some tangential comments in the intro-

duction (that seem to be unrelated to the contents of the paper).(2) The same applies to the Einstein-Dirac structures (which are not defined in the paper; presum-

ably it refers to a given solution of the Einstein-Dirac equation on the manifoldM consideredas a Spin manifold).

(3) Although the paper begins with some definitions related to Lie algebroids and Clifford al-gebroids, the connection between these and the contents of the paper is not as close as thetitle suggests. The paper’s main content is a series of computations on a manifold with a non-linear connection, in which the Lie algebroid structure plays no role, and some comments areeventually added on how to modify these results in order to make them compatible with thepresence of the anchor map.

(4) In the conclusions of the paper the author claims to have given “an intrinsic formulation ofthe geometry of CliffordN -anholonomic structures”. However, there is definitely nothingintrinsic in the paper: all the computations, definitions, etc., are given in coordinates and, bythe way, using a notation which makes the reading of the paper a challenging task.

Reviewed byJose A. Vallejo(San Luis Potosı)

References

1. S. Vacaru, J. Math. Phys.46,042503 (2005).MR2131241 (2006b:83128)2. S. Vacaru, gr–qc/ 0501057.3. F. Etayo, R. Santamarıa, and S. Vacaru, J. Math. Phys.46, 032901 (2005).MR2125569

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(2006a:53025)4. A. Cannas da Silva and A. Weinstein,Geometric Models for Noncommutative Algebras(Amer-

ican Mathematical Society, Providence, RI, 1999).MR1747916 (2001m:58013)5. R. Miron and M. Anastasiei,The Geometry of Lagrange Spaces: Theory and Applications

(Kluwer, Dordrecht, 1994).MR1281613 (95f:53120)6. S. Vacaru, E. Gaburov, and D. Gontsa, hep-th/0310133;in Clifford and Riemann- Finsler

Structures in Geometric Mechanics and Gravity, Selected Works,Differential Geometry–Dynamical Systems Monograph 7 (Geometry Balkan Press, Bucharest, 2006), Chap 2;www.mathem.pub.ro/dgds/mono/va-t.pdfand gr-qc/0508023MR2255045 (2008i:53107)

7. S. Vacaru, math.DG/0408121;Clifford and Riemann- Finsler Structures in Geometric Me-chanics and Gravity, Selected Works,Differential Geometry–Dynamical Systems Monograph7 (Geometry Balkan Press, Bucharest, 2006), Chap. 15; www.mathem.pub.ro/dgds/mono/va-t.pdfand gr-qc/0508023MR2255045 (2008i:53107)

8. S. Vacaru and O. Tintareanu-Mircea, Nucl. Phys. B626, 239 (2002). MR1891652(2004c:83111)

9. S. Vacaru and F. C. Popa, Class. Quantum Grav.18,4921 (2001).MR1869374 (2003b:83123)10. S. Vacaru and N. Vicol, Int. J. Math. Math. Sci.23,1189 (2004).MR2085061 (2005g:53135)11. S. Vacaru, J. Math. Phys.37,508 (1996).MR1370190 (96m:53085)12. S. Vacaru, J. High Energy Phys.9809,011 (1998).MR1650198 (99h:83049)13. S. Vacaru and P. Stavrinos,Spinors and Space-Time Anisotropy(Athens University Press,

Athens, Greece, 2002), gr-qc/0112028.14. A. Weinstein, Curr. Appl. Phys.7, 201 (1996).15. P. Libermann, Arch. Math.32,147 (1996).MR1421852 (98d:58203)16. E. Martinez, Acta Appl. Math.67,295 (2001).MR1861135 (2002g:37092)17. M. de Leon, J. C. Marrero, and E. Martinez, math.DG/0407528.18. S. Vacaru, Phys. Lett. B498,74 (2001).MR1815839 (2001m:81309)19. R. Penrose and W. Rindler,Two-Spinor Calculus and Relativistic FieldsSpinors and Space-

Time Vol. 1 (Cambridge University Press, Cambridge, 1984).MR0776784 (86h:83002)20. R. Penrose and W. Rindler,Spinor and Twistor Methods in Space-Time GeometrySpinors and

Space-Time Vol. 2 (Cambridge University Press, Cambridge, 1986).MR0838301 (88b:83003)21. C. P. Luehr and M. Rosenbaum, J. Math. Phys.15,1120 (1974).MR0381627 (52 #2519)22. A. Salam and J. Strathdee, Ann. Phys. (N.Y.)141,316 (1982).MR0673985 (83m:83053)23. J. M. Overduin and P. S. Wesson, Phys. Rep.283,303 (1997).MR1450560 (98g:83096)24. J. M. Gracia–Bondia, J. C. Varilli, and H. Figueroa,Elements of Noncommutative Geometry

(Birkhauser, Boston, 2001).MR1789831 (2001h:58038)25. H. Schroeder, math. DG/ 0005239.26. A. Ashtekar, Phys. Rev. D36,1587 (1987).MR0909667 (88j:83005)27. J. Kern, Arch. Math.25,438 (1974).MR0358615 (50 #11075)

Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

c© Copyright American Mathematical Society 2007, 2011

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Book

Citations

From References: 14From Reviews: 1

MR2255045 (2008i:53107)53C60 (70G45 81R60 83F05)

Vacaru, S.(E-CSIC); Stavrinos, P.; Gaburov, E.; Gonta, D.FClifford and RiemannFinsler structures in geometric mechanics and gravity. [Clifford and

Riemann-Finsler structures in geometric mechanics and gravity]Selected works.With a preface by Mihai Anastasiei.Available electronically at http://vectron.mathem.pub.ro/dgds/mono/va-t.pdf.DGDS. Differential Geometry—Dynamical Systems. Monographs, 7.Geometry Balkan Press, Bucharest,2006.l+643pp.

This work is a very difficult one to review. The format is that of a book, but actually it is acollection of 15 preprints uploaded mainly to the hep-th section of the arXiv. At first, I tried hard tomake comments for each “chapter” (i.e., each preprint), but later I became convinced that this wasabsolutely inappropriate, for me (it is a very time-consuming task) and for the reader (confrontedto 25–30 pages of comments). However, it is also complicated to make a consistent review for thewhole collection, as some of the topics treated are unrelated to each other.

At the end, the best solution I have found is to indicate the arXiv preprint to which each chaptercorresponds, so the reader can readily find it and take a look at its abstract, and to make a briefoutline of the layout and contents of the book.

In chapters 1 [S. I. Vacaru, “Generalized Finsler geometry in Einstein, string and metric-affinegravity”, preprint, http://arxiv.org/abs/hep-th/0310132] and 2 [S. I. Vacaru, E. Gaburov and D.Gontsa, “A method of constructing off-diagonal solutions in metric-affine and string gravity”,preprint, arxiv.org/abs/hep-th/0310133], the authors make a review of the theory of nonlinearconnections, introducing their own notations and rewriting some known results. A nonlinearconnection on the tangent bundlep:TM → M is a smooth splitting of the exact sequence ofvector bundles

[0→ vTM → TTMp′→ p−1TM → 0],

wherep−1TM is the pullback ofTM , andp′ is defined by factoring the tangent mapp∗:TTM →TM throughp−1TM andvTM = ker p′. For these, the usual notions of curvature, scalar cur-vature, Ricci tensor, etc., can be constructed, so it makes sense to explore what the meaning ofEinstein’s equations will be in this setting. This is done by the authors in several places of thebook, but notably in the first two chapters.

Chapter 3 [S. I. Vacaru and E. Gaburov, “Noncommutative symmetries and stability of blackellipsoids in metric-affine and string gravity”, preprint, arxiv.org/abs/hep-th/0310134] is devotedto the study of exact solutions to the aforementioned Einstein equations, mimicking the knownmethods for the case of a linear connection. In chapters 4 to 12 (respectively: [S. I. Vacaru, “Locallyanisotropic black holes in Einstein gravity”, preprint, arxiv.org/abs/gr-qc/0001020; S. I. Vacaru,P. Stavrinos and E. Gaburov, “Anholonomic triads and new classes of(2 + 1)-dimensional black

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hole solutions”, preprint, arxiv.org/abs/gr-qc/0106068; S. I. Vacaru, P. Stavrinos and D. Gontsa,“Anholonomic frames and thermodynamic geometry of 3D black holes”, preprint, arxiv.org/abs/gr-qc/0106069; S. I. Vacaru, “Off-diagonal 5D metrics and mass hierarchies with anisotropiesand running constants”, preprint, arxiv.org/abs/hep-ph/0106268; S. I. Vacaru and E. Gaburov,“Anisotropic black holes in Einstein and brane gravity”, preprint, arxiv.org/abs/hep-th/0108065; S.I. Vacaru and D. Gontsa,“Off-diagonal metrics and anisotropic brane inflation”, preprint, arxiv.org/abs/hep-th/0109114; S. I. Vacaru, “A new method of constructing black hole solutions in Einsteinand 5D gravity”, preprint, arxiv.org/abs/hep-th/0110250; “Black tori solutions in Einstein and5D gravity”, preprint, arxiv.org/abs/hep-th/0110284; “Ellipsoidal black hole-black tori systemsin 4D gravity”, preprint, arxiv.org/abs/hep-th/0111166]), which are the core of the book, theauthors develop their own method to find such exact solutions, called the “anholonomic framemethod”, and apply it to a wide variety of topics including: 3D black holes, warped and toroidalconfigurations, or some locally anisotropic models appearing in inflationary cosmology.

Chapters 13 [S. I. Vacaru, “Noncommutative Finsler geometry, gauge fields and gravity”,preprint, arxiv.org/abs/math-ph/0205023], 14 [S. I. Vacaru, “(Non)commutative Finsler geometryfrom string/M-theory”, preprint, arxiv.org/abs/hep-th/0211068] and 15 [S. I. Vacaru, “Nonholo-nomic Clifford structures and noncommutative Riemann-Finsler geometry”, preprint, arxiv.org/abs/math/0408121] are of a somewhat different nature than the rest of the papers and deal withthe generalization of the notion of a nonlinear connection to the case of noncommutative spaces.To this end, use is made of the correspondence established by the Serre-Swan theorem: the cate-gory of smooth vector bundles over a smooth manifoldM is equivalent to the category of finitelygenerated projective modules overC∞(M). Using this theorem as a guide, in noncommutativegeometry one treats finitely generated projective modules over a noncommutative algebraA asvector bundles over a suitable noncommutative space.

The authors extend the definition of nonlinear connection to finitely generated projective mod-ules, and then apply it to the study of gravity in this new context.

Reviewed byJose A. Vallejo(San Luis Potosı)

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Citations

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MR2131241 (2006b:83128)83C65Vacaru, Sergiu I. (E-CSIC)Exact solutions with noncommutative symmetries in Einstein and gauge gravity. (Englishsummary)J. Math. Phys.46 (2005),no. 4,042503, 47pp.

The author proposes to search for solutions of generalized theories of gravity that possess so-called noncommutative symmetries which in some limits may reduce to standard Killing sym-

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metries (isometries). The space-time is a fibre bundle with ann-dimensional base with a pseudo-Riemannian metricgij(xk) and, on the bundle space, is parametrized by(xk, ya) coordinates, witha metric which is a direct product ofg and a fibre component of the formhab(xk, y)δya ⊗ δyb,whereδya = dya +Na

i (x, y)dxi. This anholonomic coframe{δya} seems to incorporate the con-nection to noncommutative geometry. A (not necessarily symmetric) linear connection adaptedto this coframe can be introduced which defines the curvature. Since the vector fields dual tothe coframes{δya} do not commute they do not form, in general, a Lie algebra but are here in-terpreted as some kind of anholonomic noncommutative symmetries. The bulk of the paper isthen concerned with taking well-known (4-dimensional) solutions of Einstein’s equations (like theReissner-Nordstrom and Kerr metrics) and deforming them by adding one or more parameters toconstruct higher-dimensional metrics with noncommutative symmetries.

Reviewed byHans-Peter Kunzle

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(2002).58. H. Nishino and S. Rajpoot, hep-th/0212329.59. H. Garcia-Compean, O. Obregon, C. Ramirez, and M. Sabido, Phys. Rev. D68, 044015 (2003).

MR2035671 (2005e:83059)60. M. A. Cardella and D. Zanon, Class. Quantum Grav.20, L95 (2003). MR1979062

(2004d:83045)61. S. I. Vacaru, I. A. Chiosa, Nadejdo. A. Vicol, inNATO Advanced Research Workshop Proceed-

ings, Noncommutative Structures in Mathematics and Physics, edited by S. Duplij and J. Wess,September 23–27, Kyiv, Ukraine (Kluwer Academic Publishers, The Netherlands, 2001), pp.229–243, hep-th/0011221.MR1893448 (2002k:81009)

62. B. Jurco and P. Schupp, Eur. Phys. J. C14, 367 (2000).MR1827479 (2002f:81140)63. H. Weyl, Z. Phys.46, 1 (1927).64. H. Weyl,The Theory of Groups and Quantum Mechanics(Dover, New York, 1931); translated

from Gruppentheorie and Quantenmechanik(Hirzel Verlag, Leipzig, 1928).65. E. P. Wigner, Phys. Rev.40, 749 (1932).66. J. E. Moyal, Proc. Cambridge Philos. Soc.45, 99 (1949).MR0029330 (10,582d)67. F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, Ann. Phys. (N.Y.)111,

61 (1978).MR0496157 (58 #14737a)68. M. Kontsevich, q-alg/9709040.69. A. Zotov, Mod. Phys. Lett. A16, 615 (2001).MR1833118 (2002e:81096)70. J. Madore, S. Schraml, P. Schupp, and J. Wess, Eur. Phys. J. C16, 161 (2000).MR1826068

(2002f:81141)71. J. Wess and B. Zumino, Nucl. Phys. B, Proc. Suppl.18B, 302 (1991).MR1128150 (92g:58011)72. J. Wess, ”q-deformed Heisenberg algebras,” inProceeding of the 38 International Univer-

sitatswochen fur Kern-und Teilchenphysik, Lecture Notes in Physics, No. 543 (Springer-Verlag,Berlin, 2000), Schladming, January 1999, edited by H. Gusterer, H. Grosse, and L. Pitner, math-ph/9910013.MR1770713 (2001g:81127)

73. S. Vacaru and Yu. Goncharenko, Int. J. Theor. Phys.34, 1955 (1995).MR1351356 (96i:53068)74. S. Vacaru, hep-th/9810229.75. S. Vacaru, Buletinul Academiei de Stiinte a Republicii Moldova, Fizica si Tehnica [Izv. Acad.

Nauk Resp. Moldova, fizica i tehnika]3, 26 (1996).76. R. Utiyama, Phys. Rev.101, 1597 (1956).MR0078223 (17,1163e)77. F. Hehl, J. D. McGrea, E. W. Mielke, and Y. Ne’eman, Phys. Rep.258, 1 (1995).MR1340371

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79. L. Bonora, M. Schnabl, M. M. Sheikh-Jabbari, and A. Tomasiello, Nucl. Phys. B589, 461(2000).MR1788810 (2001h:81255)

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(2002c:81094)Note: This list reflects references listed in the original paper as accurately as possible with no

attempt to correct errors.

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From References: 6From Reviews: 0

MR2125569 (2006a:53025)53C15 (37J60 53C80 83C15)

Etayo, Fernando(E-SANTS-CP);Santamaria, Rafael[Santamarıa Sanchez, Rafael](E-SANTS-CP);Vacaru, Sergiu I. (E-SANTS-CP)Lagrange-Fedosov nonholonomic manifolds. (English summary)J. Math. Phys.46 (2005),no. 3,032901, 17pp.

A unified approach to the geometry of Lagrange and Finsler spaces and to the construction of exactsolutions with generic off-diagonal terms of the vacuum Einstein equations is presented within theframework of the theory of manifolds with anN -anholonomic structure. By the latter is meanta manifold with a nonlinear connection, brieflyN -connection, and an adapted nonholonomic (or

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anholonomic) frame. This paper, which is a continuation of recent work by the third author, ismainly rooted in the tradition of the Romanian school on Finsler geometry and the geometry ofLagrange and Hamilton spaces (cf. previous work on the subject by R. Miron, M. Anastasei andothers).

Consider a LagrangianL, defined and continuous on a tangent bundleTM , and assumeL issmooth and regular onTM := TM r {0} (i.e. the tangent bundle with the zero section deleted).From previous work it is recalled that such a Lagrangian induces a nonlinear connection and analmost Kahler structure, with associated almost symplectic form, onTM . Next, it is shown howN -anholonomic structures make their appearance in the context of gravity theories.

The notion of “distinguished connection” (d-connection) on anN -anholonomic manifold isintroduced: it is a linear connection whose parallel transport preserves the decomposition of thetangent bundle induced by the given nonholonomic connection. The existence and properties of ad-connection “compatible” with the structures induced by a regular Lagrangian are discussed.

A FedosovN -anholonomic manifold is defined as anN -anholonomic manifold with an almostcomplex and an almost symplectic structure, adapted to the givenN -connection, and an almostsymplecticd-connection. If the given structures are induced by a Lagrangian function, the manifoldis called a Lagrange-Fedosov manifold. Some properties of these manifolds are investigated.

Finally, conditions are given under which gravitational vacuum configurations withN -anholonomic structures represent exact solutions of the Einstein equations.

Reviewed byFrans Cantrijn

References

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7. R. Miron and M. Anastasiei,The Geometry of Lagrange Spaces: Theory and Applications(Kluwer, Dordrecht, 1994).MR1281613 (95f:53120)

8. R. Miron, D. Hrimiuc, H. Shimada, and V. S. Sabau,The Geometry of Hamilton and LagrangeSpaces(Kluwer, Dordrecht, 2000).MR1839409 (2002e:53111)

9. S. Vacaru, J. High Energy Phys.04, 009 (2001).10. S. Vacaru, Ann. Phys. (N.Y.)290, 83 (2001).MR1834616 (2002d:82064)11. S. Vacaru and O. Tintareanu-Mircea, Nucl. Phys. B626, 239 (2002). MR1891652

(2004c:83111)12. S. Vacaru and D. Singleton, Class. Quantum Grav.19, 2793 (2002).MR1911310 (2003c:83127)13. S. Vacaru and H. Dehnen, Gen. Relativ. Gravit.35, 209 (2003).MR1964039 (2004d:83060)14. I. Vaisman,Lectures on the Geometry of Poisson Manifolds(Birkhauser, 1994).MR1269545

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(95h:58057)15. H. C. Lee, Am. J. Math.65, 433 (1943).MR0008495 (5,15h)16. P. Libermann, Ann. Mat. Pura Appl.36, 27 (1954).MR0066020 (16,520c)17. Yu. I. Levin, Dokl. Akad. Nauk SSSR128, 668 (1959). [in Russian].MR0115142 (22 #5944)18. P. Tondeur, Comment. Math. Helv.36, 234 (1961).MR0138068 (25 #1515)19. I. Vaisman, Monatsh. Math.100, 299 (1985).MR0814206 (87d:53077)20. D. Leites, E. Poletaeva, and V. Serganova, J. Nonlinear Math. Phys.9, 394 (2002).MR1931997

(2003i:53037)21. D. Leites, unpublished, math. RT/0202213.22. A. M. Vershik, Classical and Nonclassical Dynamics with Constraints, inGeometry and Topol-

ogy in Global Nonlinear Problems, edited by Yu. G. Borisovish and Yu. F. Gliklih (Novoe Glob.Anal. Voronezh Gos. Univ. Voronezh, 1984); Lect. Notes Math.1108, 278 (1984).MR0768601(86g:58067)

23. R. Miron, Part. II. Math. Inform.50, 93 (2001).MR1981828 (2004c:53120)24. R. Miron, Algebras, Groups Geom.17, 283 (2001).MR1814931 (2001m:53131)25. R. Miron, An. Stiint. Univ. ”Al. l. Cuza” lasi, Sect. 13, 171 (1957).26. R. Miron, Acad. R. P. Romıne. Fil. Iasi Stud. Cerc. Sti. Mat.8, 49 (1957).MR0100976 (20

#7400b)27. R. Miron, An. Stiint. Univ. ”Al. l. Cuza” Iasi, Sect. 12, 85 (1956).28. V. Cruceanu, Acad. R. P. Romıne. Fil. Iasi Stud. Cerc. Sti. Mat.11, 343 (1960).MR0131238

(24 #A1091)29. F. Cantrijn, M. de Leon, J. C. Marrero, and D. Martın de Diego, Nonlinearity13, 1379 (2000).

MR1767965 (2001g:37117)30. S. Vacaru, hep-th/0310132.31. S. Vacaru, E. Gaburov and D. Gontsa, hep-th/0310133.32. S. Vacaru and E. Gaburov, hep-th/0310134.33. E. Cartan,Les Espaces de Finsler(Herman, Paris, 1935).34. A. Kawaguchi, Akad. Wetensch. Amsterdam Proc.40, 596 (1937).35. A. Kawaguchi, Tensor2, 123 (1952).MR0052188 (14,585f)36. A. Kawaguchi, Tensor6, 165 (1956).MR0084829 (18,931b)37. C. Ehresmann, Colloque de Topologie (espaces fibres), Bruxelles1950, 29 (1951).MR0042768

(13,159e)38. W. Barthel, J. Reine Angew. Math.212, 120 (1963).MR0149409 (26 #6898)39. M. de Leon and C. Villaverde, C. R. Seances Acad. Sci., Ser. 2,293, 51 (1981).MR0633561

(82j:53056)40. M. de Leon, J. Marın-Solano, and J. Marrero, An. Fis. Monogr. CIEMAT (Madrid)2, 73 (1995).

MR1357424 (96h:58064)41. F. Etayo, Riv. Mat. Univ. Parma17, 131 (1991).MR1174941 (93f:55013)42. A. Ferrandez, Rev. Roum. Math. Pures Appl.29, 225 (1984).MR0743225 (85i:53041)43. S. Vacaru, J. Math. Phys.37, 508 (1996).MR1370190 (96m:53085)44. S. Vacaru and N. Vicol, Int. J. Math. Math. Sci.23, 1189 (2004).MR2085061 (2005g:53135)45. S. Vacaru, Nucl. Phys. B434, 590 (1997).MR1453774 (98j:81303)

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46. S. Vacaru,Interactions, Strings and Isotopies in Higher Order Anisotropic Superspaces(Hadronic Press, Palm Harbor, FL, 1998).MR1679228 (2000c:81343)

47. S. Vacaru, Phys. Lett. B498, 74 (2001).MR1815839 (2001m:81309)48. J. Grifone, Ann. Inst. Fourier22, 291 (1972).MR0341361 (49 #6112)49. F. Etayo, Rev. Acad. Canaria Cienc.5, 125 (1993).MR1294582 (95h:53022)50. K. Yano and S. Ishihara,Tangent and cotangent bundles. Differential geometry(Marcel Dekker,

Inc., New York, 1973).MR0350650 (50 #3142)51. M. Crampin, Math. Proc. Cambridge Philos. Soc.94, 125 (1983).MR0704806 (84i:58009)52. J. Kern, Arch. Math.25, 438 (1974).MR0358615 (50 #11075)53. P. Finsler (Dissertation Gottingen, 1918, reprinted Basel, Birkhauser, 1951).54. H. Rund,The Differential Geometry of Finsler Spaces(Springer, Berlin, 1959).MR0105726

(21 #4462)55. G. S. Asanov,Finsler Geometry, Relativity and Gauge Theories(Kluwer, Reidel, 1985).

MR0827217 (87d:53122)56. M. Matsumoto,Foundations of Finsler Geometry and Special Finsler Spaces(Kaisisha, Shi-

gaken, 1986).MR0858830 (88f:53111)57. A. Bejancu,Finsler Geometry and Applications(Horwood, Chichester, 1990).MR1071171

(91i:53075)58. D. Bao, S.-S. Chern, and Z. Shen,An Introduction to Riemann-Finsler Geometry(Springer,

New York, 2000).MR1747675 (2001g:53130)59. Z. Shen,Differential Geometry of Spray and Finsler Spaces(Kluwer, Dordrecht, 2001).

MR1967666 (2003k:53090)60. R. Santamarıa, Ph. D. thesis, Universidad de Cantabria, Santander, Spain, 2002.61. B. Ammann, R. Lauter, and V. Nistor, Int. J. Math. Math. Sci.4, 161 (2004).MR2038804

(2005h:53050)Note: This list reflects references listed in the original paper as accurately as possible with no

attempt to correct errors.

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Citations

From References: 0From Reviews: 1

MR2117522 (2005k:58008)58A50 (53B40)

Vacaru, Sergiu I. (1-CAS2-P); Vicol, Nadejda A. (MO-MOMI)Generalized Finsler superspaces. (English summary)Conference “Applied Differential Geometry: General Relativity”—Workshop “Global Analysis,Differential Geometry, Lie Algebras”,197–229,BSG Proc., 11,Geom.Balkan Press, Bucharest,2004.

The authors study the construction of nonlinear connections and metric structures on supervector

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bundles within the framework of DeWitt-Rogers supermanifolds. These structures include thesupersymmetric version of Lagrange and Finsler spaces, and are used in order to deal with the so-called locally anisotropic spaces, which the authors consider lie at the basis of quantum statisticaland field theories of non-homogeneous and dispersive media. Supergravitational theories on theselocally anisotropic spaces (developed in a way parallel to the classical theory of Einstein as well asin a gauge-like form) are analyzed, and some comments on physical applications to quantum fieldtheory are included. From a mathematical point of view, the treatment is local, and nearly all thecomputations are done in an adapted reference frame, in concordance with the notations commonin physics.{For the entire collection seeMR2115820 (2005i:53001)}

Reviewed byJose A. Vallejo(San Luis Potosı)

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Citations

From References: 0From Reviews: 1

MR2117521 (2006e:53047)53B40Tsagas, Grigorios(GR-THESS-DM); Vacaru, Sergiu I. (1-CAS2-P)Nonlinear connections and isotopic Clifford structures. (English summary)Conference “Applied Differential Geometry: General Relativity”—Workshop “Global Analysis,Differential Geometry, Lie Algebras”,153–196,BSG Proc., 11,Geom.Balkan Press, Bucharest,2004.

The authors study nonlinear connections and Clifford structures, including spinors, curvature,torsion, etc., on (pseudo-Riemannian) isospaces. An isospace is a mutation of an algebra by aninvertible element; such spaces can be used to describe moving objects having internal degreesof freedom and lead to generalized Lagrange and Finsler structures. The authors isoify the wholebundle toolkit and finally introduce iso-Clifford algebras, isospinors using nonlinear connections.{For the entire collection seeMR2115820 (2005i:53001)}

Reviewed byBertfried Fauser

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MR2085061 (2005g:53135)53C60 (53C27 83C60)

Vacaru, Sergiu I. (P-TULT-AP); Vicol, Nadejda A. (MO-MOMI)Nonlinear connections and spinor geometry. (English summary)Int. J. Math. Math. Sci.2004,no. 21-24,1189–1237.

First, the authors develop the geometry of the total spaceE of a fiber bundleπ:E → M whosetypical fiber is a finite direct sum of vector and covector spaces. The lines of this developmentstem from the book by R. Miron and M. Anastasiei [The geometry of Lagrange spaces: theoryand applications, Kluwer Acad. Publ., Dordrecht, 1994;MR1281613 (95f:53120)]. The main toolis a nonlinear connection given as a distributon that is supplementary to the vertical distributiondefined by the kernel of the differentialπ∗ of π. Then all the geometrical objects are decomposedwith respect to these distributions. The resulting components, usually written in an adapted basisthat is anholonomic, are calledd-objects. Great attention is paid to the anholonomic frame for itsusefulness in Physics. The lineard-connections,d-metrical structures, compatibility,d-torsions,andd-curvatures are treated in detail and particular cases such as generalised Lagrange spaces,Lagrange spaces, Finsler spaces, higher order Lagrange and Hamilton spaces are reviewed.

Secondly, the authors use similar ideas for discussing spinor geometry in the spirit of Finsler-like geometries. Now the geometric support is the total space of a fiber bundle whose fibers are thedirect sum of Clifford algebras. Some constructions of great generality and interest such as Cliffordfibrations and almost complex spinor structures are provided. The decomposition produced by anonlinear connection leads to Cliffordd-algebras,d-spinors andd-twistors. A covariant derivativefor d-spinors is introduced, and, accordingly, torsions and curvatures.

In a long introduction as well as throughout the paper, many strong arguments for applica-tions of Finsler-like geometries in string and gravity theory, noncommutative geometry and thenoncommutative theory of fields and gravity are provided.

The references contains 113 titles. Almost 40 belong to the first author and his collaborators.Reviewed byMihai Anastasiei

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From References: 0From Reviews: 1

MR2077042 (2005b:83067)83C57 (83C15)

Vacaru, Sergiu I. (P-TULT-AP); Goksel, Daylan Esmer(TR-ISTUNS-P)Horizons and geodesics of black ellipsoids with anholonomic conformal symmetries.(English summary)Focus on mathematical physics research,1–14,Nova Sci.Publ., Hauppauge, NY, 2004.

Summary: “The horizon and geodesic structure of static configurations generated by anisotropicconformal transforms of the Schwarzschild metric is analyzed. We construct the maximal analytic

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extension of such off-diagonal vacuum metrics and conclude that for small deformations there aredifferent classes of vacuum solutions of the Einstein equations describing ‘black ellipsoid’ objects.This is possible because, in general, for off-diagonal metrics with deformed non-spherical sym-metries and associated anholonomic frames the conditions of the uniqueness black hole theoremsdo not hold.”{For the entire collection seeMR2073646 (2005a:00012)}

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Citations

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MR1982185 (2004f:83005)83C05 (53C80)

Dehnen, Heinz(D-KNST-P); Vacaru, Sergiu I. (P-TULT-AP)Nonlinear connections and nearly autoparallel maps in general relativity. (Englishsummary)Gen. Relativity Gravitation35 (2003),no. 5,807–850.

In this article the authors consider the metricsg = gαβduα ⊗ duβ, uα = (xk, ya), k = 1, . . . , n,a = 1, . . . ,m, on(n+m)-dimensional (pseudo) Riemannian space-timesV (n+m) that are writtenin the block diagonal form

gαβ =[gij(uα) 0

0 hab(uα)

]with respect to an anholonomic frameδα = (δi, ∂ya) whereδi = ∂xi −N b

i (xj, yc)∂yb and where the

componentsgij andhab are derived from Finsler and, more generally, Lagrange structures. Theyuse the method of moving anholonomic frames with associated nonlinear connections to examinethe conditions under which metrics of this form solve the Einstein field equations, where themetric connection is not assumed to have vanishing torsion. Using their techniques, they constructnew classes of anisotropic solutions of the Einstein field equations for both the vacuum and thenon-vacuum case. The authors also use the concept of nearly autoparallel maps to discuss localconservation laws on anisotropic spacetimes.

Reviewed byHans P. Kunzle

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MR1969737 (2004k:83076)83C57 (83C15)

Vacaru, Sergiu I. (P-TULT-AP)Horizons and geodesics of black ellipsoids. (English summary)Internat. J. Modern Phys. D12 (2003),no. 3,479–494.

Summary: “We analyze the horizon and geodesic structure of a class of 4D off-diagonal metricswith deformed spherical symmetries, which are exact solutions of the vacuum Einstein equationswith anholonomic variables. The maximal analytic extension of the ellipsoid type metrics isconstructed and the Penrose diagrams are analyzed with respect to the adapted frames. We provethat for small deformations (small eccentricities) there are metrics such that the geodesic behaviouris similar to the Schwarzcshild one. We conclude that some vacuum static and stationary ellipsoidconfigurations may describe black ellipsoid objects.”

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MR1969736 (2004k:83075)83C57 (83C25)

Vacaru, Sergiu I. (P-TULT-AP)Perturbations and stability of black ellipsoids. (English summary)Internat. J. Modern Phys. D12 (2003),no. 3,461–478.

Summary: “We study the perturbations of two classes of static black ellipsoid solutions of the four-dimensional vacuum Einstein equations. Such solutions are described by generic off-diagonalmetrics which are generated by anholonomic transforms of diagonal metrics. The analysis isperformed in the approximation of small eccentricity deformations of the Schwarzschild solution.We conclude that such anisotropic black hole objects may be stable with respect to the perturbationsparametrized by the Schrodinger equations in the framework of the one-dimensional inversescattering theory.”

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MR1964039 (2004d:83060)83C99 (53C80 83E15)

Vacaru, Sergiu I. (P-TULT-AP); Dehnen, Heinz(D-KNST-P)Locally anisotropic structures and nonlinear connections in Einstein and gauge gravity.(English summary)Gen. Relativity Gravitation35 (2003),no. 2,209–250.

In the article under review the authors begin by considering metricsg = gαβduα ⊗ duβ, uα =(xk, ya), k = 1, . . . , n, a = 1, . . . ,m, on(n+m)-dimensional (pseudo-)Riemannian space-timesV (n+m) that can be written in block diagonal form

gαβ =[

gij(xk, ya) 00 hab(xk, ya)

]with respect to a frameδα = (δi, ∂ya) whereδi = ∂xi −N b

i (xj, yc)∂yb. They refer to frames of

this type as “anholonomic”. The anholonomic frame induces a splitting of the tangent space intoa horizontal and vertical subspace. The resulting decomposition of a linear connection and theassociated curvature tensor are analyzed by the authors and all relevant formulas for the variouscomponents are provided.

Next, the authors consider an inductive generalization of the anholonomic frame which splits themetric into two or more block diagonal pieces. The frame induces a corresponding decompositionof the tangent space into a direct sum of subspaces. The authors analyze the corresponding de-composition of a linear connection and the associated curvature tensor again providing formulasfor all the components. These formulas are subsequently used to describe the resulting decompo-sition in the field equations of general relativity, affine-Poincare and/or de Sitter gauge gravity andKaluza-Klein theories.

Finally, the authors use their construction to find new classes of cosmological solutions in generalrelativity. These solutions describe Friedmann-Robertson-Walker like universes with resolutionellipsoid or torus symmetry.

Reviewed byTodd A. Oliynyk

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MR1922140 (2003k:83122)83E15 (83C57)

Vacaru, Sergiu I. (1-CAS2-P); Singleton, D.[Singleton, Douglas](1-CAS2-P)Warped solitonic deformations and propagation of black holes in 5D vacuum gravity.(English summary)Classical Quantum Gravity19 (2002),no. 14,3583–3601.

Summary: “In this paper we use the anholonomic frames method to construct exact solutions forvacuum 5D gravity with metrics having off-diagonal components. The solutions are, in general,

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anisotropic and possess interesting features such as an anisotropic warp factor with respect to theextra dimension, or a gravitational scaling/running of some of the physical parameters associatedwith the solutions. A certain class of solutions is found to describe Schwarzschild black holeswhich ‘solitonically’ propagate in spacetime. The solitonic character of these black-hole solutionsarises from the embedding of the sine-Gordon soliton configuration into certain ansatz functionsof the 5D metric. These solitonic solutions may either violate or preserve local Lorentz invariance.In addition, there is a connection between these solutions and non-commutative field theory. Inaddition to the possible physical applications of the solutions presented here, this paper is meantto illustrate the strength of the anholonomic frames method in handling anisotropic solutions ofthe gravitational field equations.”

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MR1911310 (2003c:83127)83E15Vacaru, Sergiu I. (1-CAS2-P); Singleton, D.[Singleton, Douglas](1-CAS2-P)Warped, anisotropic wormhole/soliton configurations in vacuum 5D gravity. (Englishsummary)Classical Quantum Gravity19 (2002),no. 11,2793–2811.

Summary: “In this paper we apply the anholonomic frames method developed in previous workto construct and study anisotropic vacuum field configurations in 5D gravity. Starting with anoff-diagonal 5D metric, parametrized in terms of several ansatz functions, we show that usinganholonomic frames greatly simplifies the resulting Einstein field equations. These simplifiedequations contain an interesting freedom in that one can choose one of the ansatz functions andthen determine the remaining ansatz functions in terms of this choice. As examples we take oneof the ansatz functions to be a solitonic solution of either the Kadomtsev-Petviashvili equation orthe sine-Gordon equation. There are several interesting physical consequences of these solutions.First, a certain subclass of the solutions discussed in this paper has an exponential warp factorsimilar to that of the Randall-Sundrum model. However, the warp factor depends on more than justthe fifth coordinate. In addition the warp factor arises from anisotropic vacuum solutions ratherthan from any explicit matter. Second, the solitonic character of these solutions might allow themto be interpreted either as gravitational models for particles (i.e. analogous to the ’t Hooft-Polyakovmonopole, but in the context of gravity), or as nonlinear, anisotropic gravitational waves.”

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MR1893682 (2002m:83078)83E05 (83E15)

Vacaru, Sergiu I. (1-CAS2-P); Singleton, D.[Singleton, Douglas](1-CAS2-P)Ellipsoidal, cylindrical, bipolar and toroidal wormholes in 5D gravity. (English summary)J. Math. Phys.43 (2002),no. 5,2486–2504.

The authors generalize five-dimensional vacuum Einstein solutions which can represent worm-holes and fluxtubes. The technique used to find the solutions is called the method of anholonomicframes, and the solutions found are deformations from a spherical background. The solutions arelocally anisotropic and have off-diagonal metrics which reduce to previously known sphericallysymmetric wormhole metrics in the local isotropic limit. The off-diagonal components can berelated to the “electric” and “magnetic” charges of the solution.

Reviewed byMark D. Roberts

References

1. M. S. Morris and K. S. Thorne, Am. J. Phys.56, 395 (1988); S. Giddings and A. Strominger,Nucl. Phys. B306, 890 (1988); K. A. Bronnikov and J. C. Fabris, Gravitation Cosmol.3,67 (1997); M. Rainer and A. Zhuk, Phys. Rev. D54, 6186 (1996); M. Visser,LorentzianWormholes: From Einstein to Hawking(AIP, New York, 1995).MR0941172 (89d:83001)

2. A. Salam and J. Strathdee, Ann. Phys. (N.Y.)141, 316 (1982); R. Percacci, J. Math. Phys.24,807 (1983).MR0673985 (83m:83053)

3. A. Chodos and S. Detweiler, Gen. Relativ. Gravit.14, 879 (1982); G. Clement,ibid. 16, 131(1984).MR0670161 (84a:83030)

4. V. D. Dzhunushaliev, Izv. Vuzov Fizika,78, N6 (1993) (in Russian); Gravitation Cosmol.3,240 (1997); V. D. Dzhunushaliev, Gen. Relativ. Gravit.30, 583 (1998); V. D. Dzhunushaliev,Mod. Phys. Lett. A13, 2179 (1998); V. D. Dzhunushaliev and D. Singleton, Class. QuantumGrav.16, 973 (1999).MR1240250 (94h:83114)

5. S. Vacaru, D. Singleton, V. Botan, and D. Dotenco, Phys. Lett. B519, 249 (2001).MR1861079(2002h:83113)

6. S. Vacaru, Ann. Phys. (N.Y.)256, 39 (1997); Nucl. Phys. B434, 590 (1997); J. Math. Phys.37,508 (1996); J. High Energy Phys.09, 011 (1998); Phys. Lett. B498, 74 (2001).MR1447730(98j:83103)

7. S. Vacaru, gr-qc/0001020; J. High Energy Phys.04, 009 (2001); Ann. Phys. (N.Y.)290, 83(2001); S. Vacaru and F. C. Popa, Class. Quantum Grav.18, 4921 (2001).

8. R. Miron and M. Anastasiei,The Geometry of Lagrange Spaces: Theory and Applications(Kluwer Academic, Dordrecht, 1994).MR1281613 (95f:53120)

9. A. Kawaguchi, Akad. Wetensch. Amsterdam, Proc.40, 596 (1937).10. E. Kamke,Differential Cleichungen. Losungsmethoden und Lonsungen: I. Gewohnliche Dif-

ferentialgleichungen(Chelsea, Leipzig, 1959).MR0466672 (57 #6549)

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11. V. D. Dzhunushaliev and D. Singleton, Phys. Rev. D59, 064018 (1999); gr-qc/9807086.12. H. Liu and P. S. Wesson, Phys. Lett. B377, 420 (1996).MR1399715 (97d:83098)13. H. Liu and P. S. Wesson, Class. Quantum Grav.14, 1651 (1997).MR1461142 (98g:83095)14. A. G. Agnese, A. P. Billyard, H. Liu, and P. S. Wesson, Gen. Relativ. Gravit.31, 527 (1999).

MR167941415. J. Ponce de Leon, gr-qc/0105120.16. J. M. Overduin and P. S. Wesson,Space, Time, Matter(World Scientific, Singapore, 1999);

Phys. Rep.283, 303 (1997).17. G. A. Korn and T. M. Korn,Mathematical Handbook(McGraw-Hill, New York, 1968).

Note: This list reflects references listed in the original paper as accurately as possible with noattempt to correct errors.

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Article

Citations

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MR1891652 (2004c:83111)83E15 (53C80 83C15)

Vacaru, Sergiu I. (1-CAS2-P); Tintareanu-Mircea, OvidiuAnholonomic frames, generalized Killing equations, and anisotropic Taub-NUT spinningspaces. (English summary)Nuclear Phys. B626(2002),no. 1-2,239–264.

Summary: “In this paper, we define anisotropic extensions of Euclidean Taub-NUT spaces by us-ing anholonomic frames in (pseudo-)Riemannian spaces. With respect to coordinate frames suchspaces are described by off-diagonal metrics which could be diagonalized by corresponding an-holonomic transforms. We define the conditions under which the 5D vacuum Einstein equationshave as solutions anisotropic Taub-NUT spaces. The generalized Killing equations for the con-figuration space of anisotropically spinning particles (anisotropic spinning space) are analyzed.Simple solutions of the homogeneous part of these equations are expressed in terms of someanisotropically modified Killing-Yano tensors. The general results are applied to the case of thefour-dimensional locally anisotropic Taub-NUT manifold with Euclidean signature. We empha-size that all constructions are for (pseudo-)Riemannian spaces defined by vacuum solutions, withgeneric anisotropy, of 5D Einstein equations, the solutions being generated by applying the movingframe method.”

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MR1893467 (2003e:83047)83C65 (46L87 81T75 83E50)

Vacaru, Sergiu (MO-AOS-AP); Chiosa, Iurie (MO-MOMI) ; Vicol, Nadejda (MO-MOMI)Locally anisotropic supergravity and gauge gravity on noncommutative spaces. (Englishsummary)Noncommutative structures in mathematics and physics (Kiev, 2000),229–243,NATO Sci.Ser. IIMath.Phys.Chem., 22,Kluwer Acad.Publ., Dordrecht, 2001.

The paper consists of two parts: first it is a review of the geometry of locally anisotropic superspacesand supergravity based on the formalism of anholonomic superframes and nonlinear connections.The second part is an attempt to construct gauge gravity theories on noncommutative spaces in the∗-product formalism; here the authors follow the standard approach for noncommutative gaugetheories, applying the results of [B. Jurco et al., Eur. Phys. J. C Part. Fields17 (2000), no. 3,521–526;MR1828308 (2002c:81216)] to their setup.{For the entire collection seeMR1893448 (2002k:81009)}

Reviewed byAndrzej Sitarz

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Citations

From References: 2From Reviews: 0

MR1869374 (2003b:83123)83E15 (83C15)

Vacaru, Sergiu I. (1-CAS2-P); Popa, Florian CatalinDirac spinor waves and solitons in anisotropic Taub-NUT spaces. (English summary)Classical Quantum Gravity18 (2001),no. 22,4921–4938.

Summary: “We apply a new general method of anholonomic frames with associated nonlinearconnection structure to construct new classes of exact solutions of Einstein-Dirac equations infive-dimensional (5D) gravity. Such solutions are parametrized by off-diagonal metrics in coordi-nate (holonomic) bases or, equivalently, by diagonal metrics given with respect to some anholo-nomic frames (pentads or funfbeins satisfying corresponding constraint relations). We considertwo possibilities of generalization of the Taub-NUT metric in order to obtain vacuum solutions of5D Einstein equations with effective renormalization of constants (by higher dimension anholo-nomic gravitational interactions) having distinguished anisotropies on an angular parameter oron an extra-dimensional coordinate. The constructions are extended to solutions describing self-consistent propagations of 3D Dirac wave packets in 5D anisotropic Taub-NUT spacetimes. Weshow that such anisotropic configurations of spinor matter can induce gravitational 3D solitons

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which are solutions of Kadomtsev-Petviashvili or sine-Gordon equations.”Reviewed byDmitri Korotkin

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MR1861079 (2002h:83113)83E15Vacaru, Sergiu I.; Singleton, D.[Singleton, Douglas](1-CAS2-P);Botan, Vitalie A. (MO-MOP); Dotenco, Denis A.(MO-MOP)Locally anisotropic wormholes and flux tubes in 5D gravity. (English summary)Phys. Lett. B519(2001),no. 3-4,249–259.

Summary: “In this paper we examine a class of wormhole and flux-tube-like solutions to the5D vacuum Einstein equations. These solutions possess generic local anisotropy, and their localisotropic limit is shown to be conformally equivalent to the spherically symmetric 5D solutionsby V. D. Dzhunushaliev and D. A. Singleton [Phys. Rev. D (3)59 (1999), no. 6, 064018, 6 pp.MR1678946]. The anisotropic solutions investigated here have two physically distinct signatures:first, they can give rise to angular-dependent, anisotropic ‘electromagnetic’ interactions. Second,they can result in a gravitational running of the ‘electric’ and ‘magnetic’ charges of the solutions.This gravitational running of the electromagnetic charges is linear rather than logarithmic andcould thus serve as an indirect signal for the presence of higher dimensions. The local anisotropyof these solutions is modeled using the technique of anholonomic frames with respect to whichthe metrics are diagonalized. If holonomic coordinate frames were used, then such metrics wouldhave off-diagonal components.”

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MR1834408 (2002g:83017)83C20 (83C57)

Vacaru, Sergiu I. (D-KNST-P)Anholonomic soliton-dilaton and black hole solutions in general relativity. (Englishsummary)J. High Energy Phys.2001,no. 4,Paper9, 48pp.

From the summary: “A new method of construction of integral varieties of Einstein equations inthree-dimensional (3D) and 4D gravity is presented whereby, under corresponding redefinition

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of physical values with respect to anholonomic frames of reference with associated nonlinearconnections, the structure of gravity field equations is substantially simplified. The Belinskiı-Zakharov-Maison solitons for vacuum gravitational field equations are generalized to variouscases of two and three coordinate dependencies, local anisotropy and matter sources.”

Reviewed byDmitri Korotkin

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From References: 6From Reviews: 0

MR1834616 (2002d:82064)82C40 (53B40 53C80 83C99)

Vacaru, Sergiu I. (D-KNST-P)Locally anisotropic kinetic processes and thermodynamics in curved spaces. (Englishsummary)Ann. Physics290(2001),no. 2,83–123.

The kinetic equations for one-particle distributions with collision are formulated here on a higher-dimensional space-time model consisting of a vector bundle over a Lorentz manifold of arbitrarydimension. The vector bundle carries a nonlinear connection as well as a Hermitian structure, sothat the cases of general relativity and Finsler and Lagrange space generalizations are included asspecial cases. The kinetic equations are then formulated in terms of the moving frames adapted tothe horizontal and vertical subspaces of the tangent space of the bundle. Energy density, enthalpy,specific heats, entropy and other thermodynamic quantities can then be derived, as well as transportcoefficients. Some quite explicit formulae are obtained. Most of the expressions show a strongdependence on the number of additional (anisotropic) dimensions added to the space-time.

Reviewed byHans P. Kunzle

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From References: 4From Reviews: 0

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MR1815839 (2001m:81309)81T75 (81V17 83C65)

Vacaru, Sergiu I. (D-KNST-P)Gauge and Einstein gravity from non-abelian gauge models on noncommutative spaces.(English summary)Phys. Lett. B498(2001),no. 1-2,74–82.

Summary: “Following the formalism of enveloping algebras and star product calculus we formu-late and analyze a model of gauge gravity on noncommutative spaces and examine the conditionsof its equivalence to general relativity theory. The corresponding Seiberg-Witten maps are estab-lished, which allow the definition of the respective dynamics for a finite number of gravitationalgauge field components on noncommutative spaces.”

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MR1743920 (2001f:53157)53C80 (53C07 83C40 83E15)

Vacaru, Sergiu I. (MO-AOS-AP)Nearly autoparallel maps, tensor integral and conservation laws on locally anisotropicspaces. (English summary)Fundamental open problems in science at the end of the millennium, Vol. I–III (Beijing, 1997),67–103,Hadronic Press, Palm Harbor, FL, 1999.

In this paper a theory of nearly autoparallel (na-) maps of locally anisotropic (la-) spaces isdeveloped. As a general geometric model for an la-space the total spaceE of a vector bundleendowed with a nonlinear connection, a linear connection and a metrical structure is chosen. It isassumed that the last two objects are compatible and that both are adapted to the decompositionof the tangent bundleTE produced by the first. This geometric setting is recalled in Section 2,after an introduction, following the monograph by R. Miron and M. Anastasiei [The geometry ofLagrange spaces: theory and applications, Kluwer Acad. Publ., Dordrecht, 1994;MR1281613(95f:53120)]. A curve onE is called an na-curve if along it a coplanar 2-distributionE2 containingits tangent vector is given. Here coplanar means that any vector inE2 remains there after paralleltransport. A local one-to-one map fromE to E′ is said to be an na-map if it carries any parallelcurve in an na-curve. Then a connection inE′ is a deformation with a symmetric partP and askewsymmetric partQ of a given (auxiliary) linear connectionγ. It is shown in Section 3 thatthe na-maps are parametrised by the solutions of two systems of PDE involvingP andQ. Usingthis parametrisation a classification of na-maps into four classes is provided in Section 4, with theproof in Appendix A. Next, using the curvature and Ricci tensor ofγ a classification of reciprocalna-maps into four classes is also obtained. In Section 5 the author defines the tensor integration fora tensor defined on the tensor product of two vector bundlesE andE′ whenE′ is obtained fromE

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by a nearly conformally projective map, i.e. a chain of na-maps. A Stokes-type formula is included.Following again the monograph quoted above, the author proposes in Section 6 new forms of theconservation laws on la-spaces stressing nonzero divergence of the energy-momentum tensor. Thetensor integral is used. Next he writes the Einstein equations and the said conservation laws onthe image of an na-map. The last section contains some conclusions and new projects. The levelof generality is very high since the paper comprises many other models due to the author and hiscollaborators.{For the entire collection seeMR1743918 (2000k:00034)}

Reviewed byMihai Anastasiei

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MR1765533 (2001e:83087)83E15 (53B40 83C57 83C80 83D05 83E30)

Vacaru, Sergiu I. (MO-AOS-AP)Exact solutions in locally anisotropic gravity and strings. (English summary)Particles, fields, and gravitation(Łodz, 1998), 528–537,AIP Conf.Proc., 453,Amer. Inst.Phys.,Woodbury, NY, 1998.

Some basic results on generalized Finsler Kaluza-Klein gravity and locally anisotropic strings areoutlined. Then, the exact solutions for locally anisotropic Friedmann-Robertson-Walker (FRW)universes are investigated. The anisotropic black hole solutions in three-dimensional space-timesare also obtained. The results are extended to string theory. It is concluded that a generic anisotropyof FRW metrics could generate drastic modifications of cosmological models.{For the entire collection seeMR1765483 (2001a:81006)}

Reviewed byGheorghe Zet(Iasi)

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MR1679228 (2000c:81343)81T60 (58A50 58C50 81T30 83E30)

Vacaru, Sergiu Ion (MO-AOS-AP)FInteractions, strings and isotopies in higher order anisotropic superspaces. (English

summary)Hadronic Press Monographs in Mathematics.Hadronic Press, Inc., Palm Harbor, FL,1998.ii+449pp. ISBN1-57485-032-6

This book presents a general geometric background for the theory of field interactions, stringsand diffusion processes on spaces, superspaces and isospaces with higher order anisotropy andinhomogeneity. The first part of the book is devoted to the geometry of higher-order anisotropicsuperspaces with an extension to supergravity models. The author defines the nearly autoparal-lel tensor-integral on locally anisotropic multispaces, and discusses the problem of formulation ofconservation laws on spaces with local anisotropy. The author gives a brief introduction to the the-ory of higher order anisotropic superstrings and the theory of supersymmetric locally anisotropicstochastic processes. The second part contains an introduction to the geometry of higher-orderanisotropic spaces; the distinguishing of geometric objects byN -connection structures in suchspaces is analyzed, explicit formulas for coefficients of torsions and curvatures ofN - and d-connections are presented and the field equations for gravitational interactions with higher orderanisotropy are formulated. The author also considers the theory of stochastic differential equationsfor locally anisotropic diffusion processes. The main purpose of the third part is to formulate asynthesis of the Santilli isotheory and the approach to modeling locally anisotropic geometries andphysical models on bundle spaces provided with nonlinear connection and distinguished connec-tion and metric structures. The author continues the study of the interior, locally anisotropic andinhomogeneous gravitation by extending the iso-Riemannian spaces constructions and presentinga geometric background for the theory of isofield interactions in generalized iso-Lagrange and iso-Finsler spaces. This book addresses itself both to mathematicians and physicists, to researchersand graduate students.

Reviewed bySergey Nikolaevich Roshchupkin(Simferopol′)

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MR1650198 (99h:83049)83E15 (53C15 53C60 53C80 81T20)

Vacaru, Sergiu I. (MO-AOS-AP)Spinors and field interactions in higher order anisotropic spaces. (English summary)J. High Energy Phys.1998,no. 9,Paper11, 50pp. (electronic).

Summary: “We formulate the theory of field interactions with higher order anisotropy. The con-cepts of higher order anisotropic space and locally anisotropic space (in brief, ha-space andla-space) are introduced as general ones for various types of higher order extensions of Lagrangeand Finsler geometry and higher dimension (Kaluza-Klein type) spaces. The spinors on ha-spacesare defined in the framework of the geometry of Clifford bundles provided with compatible nonlin-ear and distinguished connections and metric structures (d-connection and d-metric). The spinordifferential geometry of ha-spaces is constructed. Some related issues connected with the physi-cal aspects of higher order anisotropic interactions for gravitational, gauge, spinor, Dirac spinorand Proca fields are discussed. Motion equations in higher order generalizations of Finsler spaces,of the above-mentioned type of fields, are defined by using bundles of linear and affine frameslocally adapted to the nonlinear connection structure.”

Reviewed byIlya Shapiro

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MR1676895 (2000i:83069)83D05 (53B40)

Vacaru, Sergiu I. (MO-AOS-AP)Studies on Santilli’s locally anisotropic and inhomogeneous isogeometries. I. Isobundles andgeneralized iso-Finsler gravity. (English summary)Algebras Groups Geom.14 (1997),no. 3,211–257.

Summary: “We generalize the geometry of Santilli’s locally anisotropic and inhomogeneousisospaces to the geometry of vector isobundles provided with nonlinear and distinguished iso-connections and isometric structures. We present, apparently for the first time, the isotopies ofLagrange, Finsler and Kaluza-Klein spaces. We also continue the study of the interior, locallyanisotropic and inhomogeneous gravitation by extending the iso-Riemannian space’s construc-tion and presenting a geometric background for the theory of isofield interactions in generalizediso-Lagrange and iso-Finsler spaces.”

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MR1649561 (99m:58207)58G32 (60J60)

Vacaru, Sergiu I. (MO-AOS-AP)Locally anisotropic stochastic processes in fiber bundles. (English summary)Proceedings of the Workshop on Global Analysis, Differential Geometry and Lie Algebras(Thessaloniki, 1995),123–140,BSG Proc., 1,Geom.Balkan Press, Bucharest, 1997.

The author extends the theory of diffusion processes on Riemannian spaces to spaces with localanisotropy (vector bundles provided with compatible nonlinear, distinguished connections andmetric structures).{For the entire collection seeMR1648025 (99e:53001)}

Reviewed byCatherine Doss-Bachelet

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MR1453774 (98j:81303) 81T30 (53C60 58A50)

Vacaru, Sergiu I. (MO-AOS-AP)Superstrings in higher order extensions of Finsler superspaces. (English summary)Nuclear Phys. B494(1997),no. 3,590–656.

From the summary: “The work proposes a general background of the theory of field interactionsand strings in spaces with higher order anisotropy.. . . The first five sections cover the higherorder anisotropic superspaces. We focus on the geometry distinguished by nonlinear connectionvector superbundles, consider different supersymmetric extensions of Finsler and Lagrange spacesand analyze the structure of basic geometric objects on such superspaces. The remaining fivesections are devoted to the theory of higher order anisotropic superstrings. In the frameworkof supersymmetric nonlinear sigma-models in Finsler extended backgrounds we prove that thelow-energy dynamics of such strings contains equations of motion for locally anisotropic fieldinteractions.. . . Finally, we note that the developed computation methods are general (in someaspects very similar to those for Einstein-Cartan-Weyl spaces which is a priority compared withother cumbersome calculations in Finsler geometry) and admit extensions to various Clifford andspinor bundles.”

Reviewed byIlya Shapiro

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Article

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MR1447730 (98j:83103) 83E30 (53C80 81T17 83E15)

Vacaru, Sergiu I. (MO-AOS-SP)Locally anisotropic gravity and strings. (English summary)Ann. Physics256(1997),no. 1,39–61.

Summary: “We present an introduction to the theory of gravity on locally anisotropic spaces mod-elled as vector bundles provided with compatible nonlinear and distinguished linear connectionand metric structures (such spaces are obtained by a nonlinear connection reduction or compactifi-cation from higher-dimensional spaces to lower-dimensional ones and contain as particular casesvarious generalizations of Kaluza-Klein and Finsler geometry). We analyze the conditions forconsistent propagation of closed strings in locally anisotropic background spaces. The connec-tion between conformal invariance, the vanishing of the renormalization groupβ-function of thegeneralizedσ-model, and field equations of locally anisotropic gravity are studied in detail.”

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MR1490996 83D05 (53B40 53B50)

Vacaru, S.Gauge like treatment of generalized Lagrange and Finsler gravity. (English, Moldaviansummaries)Izv. Akad. Nauk Respub. Moldova Fiz. Tekhn.1996,no. 3,31–35.

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MR1490995 (98k:53027)53C07 (58E15)

Vacaru, S.Yang-Mills fields on spaces with local anisotropy. (English, Moldavian summaries)Izv. Akad. Nauk Respub. Moldova Fiz. Tekhn.1996,no. 3,26–31.

Gauge field theories on spaces with local anisotropy are studied for the case of semisimple groups.The method is based on the almost Hermitian model of generalized Lagrange spacesH2n. Thesespaces contain the Lagrange and Finsler geometries as particular cases. The gauge fields associatedto a groupG are defined by means of a connection on the principal bundle(P, π, G,H2n) of baseH2n and withG as structural group. The Yang-Mills equations for the gauge fields are writtenin the total space of this bundle. Then the corresponding equations are projected onto the basespaceH2n in order to give a physical meaning of the results. It is shown that in the case whenG is semisimple the Yang-Mills equations can be obtained by variation of an appropriate actionintegral.

Reviewed byGheorghe Zet(Iasi)

c© Copyright American Mathematical Society 1998, 2011

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MR1490994 (98m:58150)58G32 (53C60 60H10)

Vacaru, S.Stochastic differential equations on spaces with local anisotropy. (English, Moldaviansummaries)Izv. Akad. Nauk Respub. Moldova Fiz. Tekhn.1996,no. 3,13–25.

The author’s aim is to formulate a theory of diffusion processes on spaces with local anisotropy.He formulates a theory of stochastic differential equations on generalized Lagrange spaces, whichcontains as particular cases Lagrange and Finsler spaces. The author uses an adapted nonlinearconnection as proposed by R. Miron and M. Anastasiei [The geometry of Lagrange spaces: theoryand applications, Kluwer Acad. Publ., Dordrecht, 1994;MR1281613 (95f:53120)].

Reviewed byJacques Vauthier

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MR1490993 (99b:53018)53A45 (53B40)

Vacaru, S. I.Tensorial integration on spaces with local anisotropy. (English, Moldavian summaries)Izv. Akad. Nauk Respub. Moldova Fiz. Tekhn.1996,no. 1,46–51.

It is shown that the formalism of tensorial integration defined for locally parallelizable Riemann-Cartan spaces [A. Moor, Acta Math.86(1951), 71–83;MR0045431 (13,583a); Monatsh. Math.70(1966), 134–148;MR0217729 (36 #818)] can be extended for generalized Lagrange (GL) spaces.Locally parallelizable GL-spaces admit a structure of tensorial integration ofd-tensors and therebyenable one to define conservation laws there.

Reviewed byLaszlo Kozma

c© Copyright American Mathematical Society 1999, 2011

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MR1403766 (97i:83047)83C60 (53C20 53C22 53C80)

Vacaru, Sergiu I. (MO-AOS-AP); Ostaf, Sergiu V.Twistors and nearly autoparallel maps. (English summary)Rep. Math. Phys.37 (1996),no. 3,309–324.

One of the major obstacles still confronting twistor theory is the lack of a natural twistor-theoreticdescription of the (local) geometry of an arbitrary curved space-time(V). One way to think oftwistors, in conformally flat space-times, is as solutions of the so-called twistor equation [cf., e.g.,R. Penrose and W. Rindler,Spinors and space-time. Vol. 2, Cambridge Univ. Press, Cambridge,1986;MR0838301 (88b:83003)]. The presence of conformal curvature, however, thwarts such anapproach for more general space-times due to a constraint analogous to the Buchdahl constraintsfor massless fields.

The idea of the authors of the paper under review is to associate toV an auxiliary (conformally)flat space-timeM and to “transport” twistors from the latter toV (locally). The machinery theyemploy for this transportation was developed earlier [e.g., N. S. Sinyukov,Geodesic mappings ofRiemannian spaces(Russian), “Nauka”, Moscow, 1979;MR0552022 (81g:53014)]; specifically,a (composition of finitely many) “nearly geodesic” mapping(s) fromV to M. This machineryeffectively translates the (conformally) flat space-time twistor equation into a spinor equation onV with a four-complex-dimensional solution space which may be taken to be the “twistor space”of V.

The choice of this machinery seems rather unmotivated, however, bearing no apparent connec-tion with twistor geometry itself. Indeed, the authors note that a null twistor from the auxiliaryM

is transported into a nonnull “twistor” ofV. The authors offer no application of their “twistors”;indeed, their introduction of these “twistors” seems rather superfluous to the geometric consider-

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ations of the paper concerning nearly geodesic mappings.It is perhaps worth remarking that the paper under review provides a good example of the need

for text editing of papers written in other than the authors’ first language. It is a sad trend that manyjournals make no apparent effort in this direction, with the result that sentences that are virtuallyincomprehensible appear in print, which is a disservice to all.

Reviewed byPeter R. Law

c© Copyright American Mathematical Society 1997, 2011

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MR1378664 (97b:53028)53B40Vacaru, S.; Ostaf, S.Nearly autoparallel maps of Lagrange and Finsler spaces.Lagrange and Finsler geometry,241–253,Fund.Theories Phys., 76,Kluwer Acad.Publ.,Dordrecht, 1996.

In a Lagrange spaceL(x, y) a curveγ in TM is called distinguished nearly autoparallel if a2-dimensional coplanar distribution containing the tangents ofγ exits alongγ. A one-to-onemapping between Lagrange spacesL and L is called nearly autoparallel when all autoparallelcurves ofL are mapped into a nearly autoparallel curve ofL. The authors find criteria for thesemappings and classify them into 4 classes.{For the entire collection seeMR1378642 (96j:53024)}

Reviewed byLaszlo Kozma

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Book

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MR1378661 (96k:53028)53B40Gottlieb, I. [Gottlieb, Ioan] ; Vacaru, S.A. Moor’s tensorial integration in generalized Lagrange spaces.Lagrange and Finsler geometry,209–216,Fund.Theories Phys., 76,Kluwer Acad.Publ.,Dordrecht, 1996.

Tensor integration as the inverse operation to covariant derivation is defined by the authors ongeneralized Lagrange spaces, i.e. on metric line-element spacesMn = (M, gij(x, y)) given by themetric tensorgij which need not be homogeneous iny. Then Stokes-type formulae and tensor-integral conservation laws are obtained.{For the entire collection seeMR1378642 (96j:53024)}

Reviewed byL. Tamassy

c© Copyright American Mathematical Society 1996, 2011

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MR1370190 (96m:53085)53C60 (53A50 53C80)

Vacaru, Sergiu I. (MO-AOS-AP)Spinor structures and nonlinear connections in vector bundles, generalized Lagrange andFinsler spaces. (English summary)J. Math. Phys.37 (1996),no. 1,508–523.

Let (E, π, M) be a vector bundle andHE be a nonlinear connection onE, i.e.,TE = HE⊕V E,whereV E is the vertical vector bundle ofE. Both distributionsHE andV E are assumed to beendowed with semi-Riemannian metrics. The main purpose of the present paper is to extend boththe Clifford and spinor structures to vector bundles equipped with the above geometric structure.Special attention is given to the caseE = TM , where the same metric is taken on bothHTM andV TM .

Reviewed byAurel Bejancu

c© Copyright American Mathematical Society 1996, 2011

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MR1490992 (98i:53022)53A50 (53B40)

Vacaru, S.Clifford structures and spinors on spaces with local anisotropy. (English, Moldaviansummaries)Izv. Akad. Nauk Respub. Moldova Fiz. Tekhn.1995,no. 3,53–62.

Summary: “We define Clifford and spinor structures on bundle spaces provided with nonlinearand distinguished connections and metric structures containing as particular cases generalizedLagrange and Finsler spaces.”

Reviewed byJayme Vaz, Jr.

c© Copyright American Mathematical Society 1998, 2011

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MR1351356 (96i:53068)53C60 (53C80 81T13 83D05)

Vacaru, Sergiu (MO-AOS-AP); Goncharenko, Yurii [Goncharenko, Yu. A.] (MO-AOS-AP)Yang-Mills fields and gauge gravity on generalized Lagrange and Finsler spaces. (Englishsummary)Internat. J. Theoret. Phys.34 (1995),no. 9,1955–1980.

The authors bring several original contributions to the Lagrangian gravity expounded in the bookby R. Miron and the reviewer [The geometry of Lagrange spaces: theory and applications, KluwerAcad. Publ., Dordrecht, 1994;MR1281613 (95f:53120)]. Thus, they formulate a geometric ap-proach to interactions of Yang-Mills fields on spaces with local anisotropy in vector bundles (withsemisimple structural groups) on generalized Lagrange spaces. They also extend the geometricformalism, including theories with non-semisimple groups which permit a unique fibre bundletreatment for both locally anisotropic Yang-Mills and gravitational interactions. One of the mostimportant results is that Lagrangian gravity is equivalent to a gaugelike theory in the bundle ofaffine adapted frames on generalized Lagrange spaces. This result is used for the constructionof solutions of gravitational field equations describing gravitational gauge instantons with localanisotropy.

Reviewed byMihai Anastasiei

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MR1342221 (96f:83050)83C45 (81V17)

Vacaru, S. I.; Goncharenko, Yu. A.On exactly solvable4D quantum gravities. (English, Russian, Moldavian summaries)Izv. Akad. Nauk Respub. Moldova Fiz. Tekhn.1994,no. 3,53–58, 141.

The notion of “nearly autoparallel maps” is introduced for four-dimensional smooth space-timemanifolds with an affine connectionΓa

bc and its properties are studied. Given an autoparallelvector fieldua (i.e. such thatub∇bu

a ∼ ua), the authors define a nearly autoparallel map as adeformation of an affine connectionΓa

bc −→ Γabc = Γa

bc + P abc where the defect tensor satisfies

the equation(∇cPdab + P d

kaPkbc)uaubuc = aud + bP d

abuaub with the coefficientsa, b being

the functions of space-time coordinates andua. Classification of such maps is suggested whichincludes four classes distinguished by the algebraic structure ofP a

bc and the form of the functionsa, b. The authors propose to use these maps for constructing the exact solutions of the vacuumEinstein equations from a gravitational field, represented by the connectionΓa

bc defined originallyon a two-dimensional submanifold. Although the consideration is purely classical, the authorsclaim that this approach is extendable to the quantum theory of gravity.

Reviewed byYu. N. Obukhov

c© Copyright American Mathematical Society 1996, 2011

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MR1342220 (96f:53094)53C60 (53C80)

Vacaru, S.; Ostaf, S.; Goncharenko, Yu.; Doina, A.Nearly autoparallel maps of Lagrange spaces. (English, Russian, Moldavian summaries)Izv. Akad. Nauk Respub. Moldova Fiz. Tekhn.1994,no. 3,42–53, 141.

A Lagrange space [c.f. J. Kern, Arch. Math. (Basel)25(1974), 438–443;MR0358615 (50 #11075)]is a pairLm = (M,L), whereM is anm-dimensional manifold andL is a regular Lagrangian. Anearly autoparallel mapping between two Lagrange spaces transforms a distinguished autoparallelcurve of the first space into a nearly distinguished autoparallel curve of the second space. Theauthors obtain a classification of nearly autoparallel mappings and propose an extension of theEinstein theory of gravity to Lagrange spaces.

Reviewed byAurel Bejancu

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MR1330660 (96c:83028)83C40Vacaru, S.(MO-AOS-AP)Nearly autoparallel maps and conservation laws on curved spaces. (English summary)Romanian J. Phys.39 (1994),no. 1,37–52.

The author considers local one-to-one mappings transforming Einstein-Cartan spaces, and char-acterizes these transformations by means of related geometric objects defined on such spaces. Inparticular, he considers the problem of conservation laws for gravitational fields. He also relatesthis study with previous ones, principally by A.Z.Petrov.

The meaning of the paper is summarized in the two last propositions, stating that the dynamicsof the Einstein gravitational and matter fields given on a pseudo-Riemannian space-time can beequivalently locally modelled on suitable flat backgrounds, and that this method allows us also toobtain the related conservation laws for Einstein graviational fields.{Reviewer’s remarks:. The method considered in this paper gives good information from the

local point of view. However, in the framework of quantum phenomena, global information isnecessary in order to describe tunneling effects. (See recent works on the geometry of PDEs andquantization of PDEs.)}

Reviewed byAgostino Prastaro

c© Copyright American Mathematical Society 1996, 2011

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MR1316048 (95m:83084)83C60 (83C45)

Vacaru, S. I.; Ostaf, S. V.Twistor quantum cosmology. (English, Russian, Moldavian summaries)Izv. Akad. Nauk Respub. Moldova Fiz. Tekhn.1994,no. 1,72–79, 98.

The work is mainly concerned with the construction of twistor solutions to the Wheeler-DeWittequation in the framework of a two-dimensional superspace model. Roughly speaking, the modelconsists of the minimal coupling of a homogeneous time-dependent uncharged scalar field withthe metric of a (bounded) Friedmann-Robertson-Walker space-time, the cosmological constantbeing effectively assumed to vanish. The relevant configuration spaces are “coordinatized” by thescalar field and a quantity which carries the scaling factor occurring in the (spatial)S3-part ofthe background metric along with the rest-mass of the field. The authors build up a variationalprinciple which particularly involves holding fixed the field on the boundary three-sphere at

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two different values of the time coordinate. They state that whenever the (measure-dependent)Hawking parameter is set equal to one, the pieces of the corresponding field equation turn outto be proportional to those of a Klein-Gordon equation which is set upon a conformally flatsuperspace. Their version of the Wheeler-DeWitt theory appears as an equation defined on theflat superspace involved, the result of the pertinent conformal transformation thus amounting toascribing a variable-mass character to the wave function of the universe. The twistor representationof the wave function is accomplished by utilizing contour integrals which are formally the same asthe universal projective ones for spinless massive fields, but the basic (non-projective) twistor spaceshows up as the direct sum of two spin fibers defined over points of the flat superspace. In writingthe integrals for the Klein-Gordon equation, they accordingly employ the conventional prescriptionfor constructing the conformal transformation laws for the spinor parts of elementary twistorstogether with the Penrose expression for the d’Alembertian operator. A quantum description ofthe universe is then achieved by simply making use of the standard twistor quantization rules. Atthis stage, they suggest a twistor-diagram representation of mass eigenstates which carries explicitHankel functions. It is pointed out that, under certain circumstances, one can consider alternativemodels which involve field equations containing constant-mass terms.

Reviewed byJorge Goncalves Cardoso

c© Copyright American Mathematical Society 1995, 2011

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MR1316047 (95k:83031)83C40 (83D05)

Vacaru, S. I.; Ostaf, S. V.Nearly geodesic tensorial integration on bispaces: formulation of conservation laws forgauge gravity. (English, Russian, Moldavian summaries)Izv. Akad. Nauk Respub. Moldova Fiz. Tekhn.1994,no. 1,64–72, 98.

The authors consider conservation laws for classical fields on curved spaces. The method is basedon a covariant variational principle and therefore naturally implies the use of an affine connection.The second ingredient is a decomposition of the affine connectionΓ in a background connectionplus a tensor, which the authors interpret as a deformation of the original connection. In this sensethe method is similar to that previously developed by other authors (Francaviglia and Ferraris,Krupka). The natural application of the method is to general relativity and to Yang-Mills fields.

Reviewed byVictor Tapia

c© Copyright American Mathematical Society 1995, 2011

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MR1321282 (95k:53034)53C07 (32L25)

Vacaru, S. I.Nearly autoparallel and twistor transforms of bundle spaces. (English, Russian, Moldaviansummaries)Izv. Akad. Nauk Respub. Moldova Fiz. Tekhn.1993,no. 3,17–29, 120.

Summary: “The theory of nearly autoparallel maps (na-maps) of bundle spaces is formulated. Weadvance a classification of the mentioned spaces by using chains of na-maps. The na-map methodsare used for constructions of na-projective solutions of the Yang-Mills and Einstein equations. Thetwistor-gauge interpretation is formulated on flat na-backgrounds.”{Reviewer’s remark: The paper is badly written, its constructions are not motivated, and its main

results (such as, e.g., Theorems 4 and 5, which claim the existence of isomorphisms betweensolution spaces of self-dual and non-self-dual Einstein equations in4 dimensions and certainCR-cohomology groups on the real quadric of null twistors inP3) are evidently wrong.}

Reviewed bySergey Merkulov

c© Copyright American Mathematical Society 1995, 2011

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MR1321281 (96d:83094)83D05 (53B50)

Vacaru, S. I.; Ostaf, S. V.Nearly autoparallel mappings in gravitational theories with torsion and nonmetricity.(English, Russian, Moldavian summaries)Izv. Akad. Nauk Respub. Moldova Fiz. Tekhn.1993,no. 3,4–16, 120.

The main purpose of this rather mathematical paper is the extension of the theory of geodesicmappings [see N. S. Sinyukov,Geodesic mappings of Riemannian spaces(Russian), “Nauka”,Moscow, 1979;MR0552022 (81g:53014)] of affine connected spaces (i.e., those with metric andnonsymmetric connection) to affine metric spaces (with metric, nonsymmetric connection andnonmetricity tensor). Unlike Sinyukov, the authors call such mappings nearly geodesic, seem-ingly because Sinyukov used the geodesic equation in an arbitrary parametrization:duµ/dη +Γµ

αβuαuβ = ρ(η)uµ whereuµ = dxµ/dη, Γµαβ is the affine connection, andρ(η) is a scalar func-

tion of the parameterη. The authors define an autoparallel curve as follows:duµ/dη +γµαβuαuβ =

ρ(η)uµ whereγµαβ is the symmetric part of the affine connection,γµ

αβ = Γµ(αβ). The definition of

a nearly autoparallel curve extends the one above and involves a coplanar two-dimensional vec-

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tor space along the curve. A mapping is called nearly autoparallel if an autoparallel curve of oneaffine metric space maps onto a nearly autoparallel curve of another affine metric space. The clas-sification of affine metric spaces based on chains of nearly autoparallel mappings is presented. Insuch spaces admitting nearly autoparallel maps, the authors consider a gravitational model whichis described by some generalized Einstein equations.

Reviewed byA. D. Popova

c© Copyright American Mathematical Society 1996, 2011

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MR1219085 (94c:83074)83C60 (83C45)

Vacaru, S.(MO-AOS-AP)Minisuperspace twistor quantum cosmology. (English summary)Studia Univ. Babes-Bolyai Phys.35 (1990),no. 2,36–42.

The author considers a minisuperspace model consisting of a homogeneous massive scalar fieldminimally coupled to a Friedmann-Robertson-Walker space-time metric for which the Wheeler-DeWitt (WD) equation is proportional to the Klein-Gordon (KG) equation for an associatedconformally flat metric. Solutions for the relevant WD equation may therefore be obtained fromsolutions of the relevant KG equation, and the author generates the latter by twistor techniques fromholomorphic functions of twistor and dual twistor variables. Such functions are not free, however,but must satisfy the “twistorial Klein-Gordon equation” [cf., e.g., R. Penrose, inQuantum gravity(Chilton, 1974), 268–407, Clarendon Press, Oxford, 1975; seeMR0459344 (56 #17537)]. Theauthor employs twistor-diagram methods to generate suitable solutions.

The author proposes to regard such twistor functions as a twistor representation of wave functionsof his model universe [J. B. Hartle and S. W. Hawking, Phys. Rev. D (3)28 (1983), no. 12, 2960–2975;MR0726732 (85i:83022)] and the usual twistor quantization rules as providing a “twistorquantum cosmology”.

Reviewed byPeter R. Law

c© Copyright American Mathematical Society 1994, 2011

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