Finsler and Lagrange Geometries in Einstein and String Gravity

June 17, 2008 14:9 WSPC/IJGMMP-J043 00289

International Journal of Geometric Methods in Modern PhysicsVol. 5, No. 4 (2008) 473–511c© World Scientific Publishing Company

FINSLER AND LAGRANGE GEOMETRIES IN EINSTEINAND STRING GRAVITY

SERGIU I. VACARU

The Fields Institute for Research in Mathematical Science222 College Street, 2d Floor, Toronto M5T 3J1, Canada

sergiu−[email protected]@fields.utoronto.ca

Received 21 July 2007Revised 23 January 2008Accepted 3 February 2008

We review the current status of Finsler–Lagrange geometry and generalizations. The goalis to aid non-experts on Finsler spaces, but physicists and geometers skilled in generalrelativity and particle theories, to understand the crucial importance of such geometricmethods for applications in modern physics. We also would like to orient mathematiciansworking in generalized Finsler and Kahler geometry and geometric mechanics how theycould perform their results in order to be accepted by the community of “orthodox”physicists.

Although the bulk of former models of Finsler–Lagrange spaces where elaboratedon tangent bundles, the surprising result advocated in our works is that such locallyanisotropic structures can be modeled equivalently on Riemann–Cartan spaces, evenas exact solutions in Einstein and/or string gravity, if nonholonomic distributions andmoving frames of references are introduced into consideration.

We also propose a canonical scheme when geometrical objects on a (pseudo) Rie-mannian space are nonholonomically deformed into generalized Lagrange, or Finsler,configurations on the same manifold. Such canonical transforms are defined by the coef-ficients of a prime metric and generate target spaces as Lagrange structures, their modelsof almost Hermitian/Kahler, or nonholonomic Riemann spaces.

Finally, we consider some classes of exact solutions in string and Einstein gravitymodeling Lagrange–Finsler structures with solitonic pp-waves and speculate on theirphysical meaning.

Keywords: Nonholonomic manifolds; Einstein spaces; string gravity; Finsler andLagrange geometry; nonlinear connections; exact solutions; Riemann–Cartan spaces.

MSC: 53B40, 53B50, 53C21, 53C55, 83C15, 83E99

PACS: 04.20.Jb, 04.40.-b, 04.50.+h, 04.90.+e, 02.40.-k

1. Introduction

The main purpose of this survey is to present an introduction to Finsler–Lagrangegeometry and the anholonomic frame method in general relativity and gravitation.

473


474 S. I. Vacaru

We review and discuss possible applications in modern physics and provide alter-native constructions in the language of the geometry of nonholonomic Riemannianmanifolds (enabled with nonintegrable distributions and preferred frame struc-tures). It will be emphasized that the approach when Finsler-like structures aremodeled in general relativity and gravity theories with metric compatible connec-tions and, in general, nontrivial torsion.

Usually, gravity and string theory physicists may remember that Finsler geom-etry is a quite “sophisticate” spacetime generalization when Riemannian metricsgij(xk) are extended to Finsler metrics gij(xk, yl) depending on both local coordi-nates xk on a manifold M and “velocities” yl on its tangent bundle TM .a Perhaps,they will say additionally that in order to describe local anisotropies dependingonly on directions given by vectors yl, the Finsler metrics should be defined in theform gij ∼ ∂F 2

∂yi∂yj , where F (xk, ζyl) = |ζ|F (xk, yl), for any real ζ = 0, is a fun-damental Finsler metric function. A number of authors analyzing possible locallyanisotropic physical effects omit a rigorous study of nonlinear connections and donot reflect on the problem of compatibility of metric and linear connection struc-tures. If a Riemannian geometry is completely stated by its metric, various modelsof Finsler spaces and generalizations are defined by three independent geometricobjects (metric and linear and nonlinear connections) which in certain canonicalcases are induced by a fundamental Finsler function F (x, y). For models with dif-ferent metric compatibility, or non-compatibility, conditions, this is a point of addi-tional geometric and physical considerations, new terminology and mathematicalconventions. Finally, a lot of physicists and mathematicians have concluded thatsuch geometries with generic local anisotropy are characterized by various types ofconnections, torsions and curvatures which do not seem to have physical meaningin modern particle theories but (maybe?) certain Finsler-like analogs of mechanicalsystems and continuous media can be constructed.

A few rigorous studies on perspectives of Finsler-like geometries in standardtheories of gravity and particle physics (see, for instance, [12, 85]) were publishedbut they do not analyze any physical effects of the nonlinear connection and adaptedlinear connection structures and the possibility to model Finsler-like spaces as exactsolutions in Einstein and sting gravity [77]). The results of such works, on Finslermodels with violations of local Lorentz symmetry and nonmetricity fields, can besummarized in a very pessimistic form: both fundamental theoretic consequencesand experimental data substantially restrict the importance for modern physics oflocally anisotropic geometries elaborated on (co) tangent bundles,b see Introduction

aWe emphasize that Finsler geometries can be alternatively modeled if yl are considered as certainnonholonomic, i.e. constrained, coordinates on a general manifold V, not only as “velocities” or“momenta”, see further constructions in this work.bIn result of such opinions, the editors and referees of some top physical journals almost stopped toaccept for publication manuscripts on Finsler gravity models. If other journals were more tolerantwith such theoretical works, they were considered to be related to certain alternative classes oftheories or to some mathematical physics problems with speculations on geometric models and“nonstandard” physics, mechanics and some applications to biology, sociology or seismology etc.


Finsler and Lagrange Geometries in Einstein and String Gravity 475

to monograph [77] and article [64] and reference therein for more detailed reviewsand discussions.

Why should we give special attention to Finsler geometry and methods andapply them in modern physics? We list here a set of counter–arguments and discussthe main sources of “anti-Finsler” skepticism which (we hope) will explain andremove the existing unfair situation where spaces with generic local anisotropy arenot considered in standard theories of physics:

1. One should emphasize that in the bulk the criticism on locally anisotropic geome-tries and applications in standard physics was motivated only for special classesof models on tangent bundles, with violation of local Lorentz symmetry (evensuch works became very important in modern physics, for instance, in relationto brane gravity [16] and quantum theories [28]) and nonmetricity fields. Not alltheories with generalized Finsler metrics and connections were elaborated in thisform (on alternative approaches, see next points) and in many cases, like [12,85],the analysis of physical consequences was performed not following the nonlinearconnection geometric formalism and a tensor calculus adapted to nonholonomicstructures which is crucial in Finsler geometry and generalizations.

2. More recently, a group of mathematicians [8, 54] developed intensively somedirections on Finsler geometry and applications following the Chern’s linearconnection formalism proposed in 1948 (this connection is with vanishing tor-sion but noncompatible with the metric structure). For non-experts in geometryand physics, the works of this group, and other authors working with gener-alized local Lorentz symmetries, created a false opinion that Finsler geometrycan be elaborated only on tangent bundles and that the Chern connection isthe “best” Finsler generalization of the Levi–Civita connection. A number ofvery important constructions with the so-called metric compatible Cartan con-nection, or other canonical connections, were moved on the second plan andforgotten. One should emphasize that the geometric constructions with the wellknown Chern or Berwald connections cannot be related to standard theories ofphysics because they contain nonmetricity fields. The issue of nonmetricity wasstudied in detail in a number of works on metric-affine gravity, see review [22]and Chapter I in the collection of works [77], the last one containing a series ofpapers on generalized Finsler–affine spaces. Such results are not widely acceptedby physicists because of absence of experimental evidences and theoretical com-plexity of geometric constructions. Here we note that it is a quite sophisti-cated task to elaborate spinor versions, supersymmetric and noncommutativegeneralizations of Finsler-like geometries if we work with metric noncompatibleconnections.

3. A non-expert in special directions of differential geometry and geometric mechan-ics, may not know that beginning Cartan (1935) [15] various models of Finslergeometry were developed alternatively by using metric compatible connectionswhich resulted in generalizations to the geometry of Lagrange and Hamiltonmechanics and their higher order extensions. Such works and monographs were


476 S. I. Vacaru

published by prominent schools and authors on Finsler geometry and generaliza-tions from Romania and Japan [10,11,24–26,33–39,41,42,44,45,57,84] followingapproaches quite different from the geometry of symplectic mechanics and gen-eralizations [29–32]. As a matter of principle, all geometric constructions withthe Chern and/or simplectic connections can be redefined equivalently for metriccompatible geometries, but the philosophy, aims, mathematical formalism andphysical consequences are very different for different approaches and the particlephysics researches are usually not familiar with such results.

4. It should be noted that for a number of scientists working in Western Countriesthere are less known the results on the geometry of nonholonomic manifoldspublished in a series of monographs and articles by Vranceanu (1926), Horak(1927) and others [23,80–82], see historical remarks and bibliography in [11,77].The importance for modern physics of such works follows from the idea andexplicit proofs (in quite sophisticate component forms) that various types oflocally anisotropic geometries and physical interactions can be modeled on usualRiemannian manifolds by considering nonholonomic distributions and holonomicfibrations enabled with certain classes of special connections.

5. In our works (see, for instance, reviews and monographs [58–62, 64, 76–79],and references therein), we reoriented the research on Finsler spaces and gen-eralizations in some directions connected to standard models of physics andgauge, supersymmetric and noncommutative extensions of gravity. Our basicidea was that both the Riemann–Cartan and generalized Finsler–Lagrangegeometries can be modeled in a unified manner by corresponding geometricstructures on nonholonomic manifolds. It was emphasized, that prescribing apreferred nonholonomic frame structure (equivalently, a nonintegrable distri-bution with associated nonlinear connection) on a manifold, or on a vectorbundle, it is possible to work equivalently both with the Levi–Civita and theso-called canonical distinguished connection. We provided a number of exam-ples when Finsler–like structures and geometries can be modeled as exact solu-tions in Einstein and string gravity and proved that certain geometric meth-ods are very important, for instance, in constructing new classes of exactsolutions.

This review work has also pedagogical scopes. We attempt to cover key aspectsand open issues of generalized Finsler–Lagrange geometry related to a consistentincorporation of nonlinear connection formalism and moving/deformation framemethods into the Einstein and string gravity and analogous models of gravity, seealso [10, 38, 53, 64, 77] for general reviews, written in the same spirit as the presentone but in a more comprehensive, or inversely, with more special purposes forms.While the article is essentially self-contained, the emphasis is on communicatingthe underlying ideas and methods and the significance of results rather than onpresenting systematic derivations and detailed proofs (these can be found in thelisted literature).



The subject of Finsler geometry and applications can be approached in differ-ent ways. We choose one of which is deeply rooted in the well established gravityphysics and also has sufficient mathematical precision to ensure that a physicistfamiliar with standard textbooks and monographs on gravity [21, 40, 56, 83] andstring theory [18, 51, 55] will be able without much efforts to understand recentresults and methods of the geometry of nonholonomic manifolds and generalizedFinsler–Lagrange spaces.

We shall use the terms “standard” and “nonstandard” models in geometry andphysics. In connection to Finsler geometry, we shall consider a model to be a stan-dard one if it contains locally anisotropic structures defined by certain nonholo-nomic distributions and adapted frames of reference on a (pseudo) Riemannian orRiemann–Cartan space (for instance, in general relativity, Kaluza–Klein theoriesand low energy string gravity models). Such constructions preserve, in general, thelocal Lorentz symmetry and they are performed with metric compatible connec-tions. The term “nonstandard” will be used for those approaches which are relatedto metric non-compatible connections and/or local Lorentz violations in Finslerspacetimes and generalizations. Sure, any standard or nonstandard model is rig-orously formulated following certain purposes in modern geometry and physics,geometric mechanics, biophysics, locally anisotropic thermodynamics and stochas-tic and kinetic processes and classical or quantum gravity theories. Perhaps, it willbe the case to distinguish the class of “almost standard” physical models withlocally anisotropic interactions when certain geometric objects from a (pseudo)Riemannian or Riemann–Cartan manifolds are lifted on a (co) tangent or vec-tor bundles and/or their supersymmetric, non-commutative, Lie algebroid, Cliffordspace, quantum group . . . generalizations. There are possible various effects with“nonstandard” corrections, for instance, violations of the local Lorentz symmetryby quantum effects but in some classical or quantum limits such theories are con-strained to correspond to certain standard ones.

This contribution is organized as follows:

In Sec. 2, we outline a unified approach to the geometry of nonholonomic distri-butions on Riemann manifolds and Finsler–Lagrange spaces. The basic conceptson nonholonomic manifolds and associated nonlinear connection structures areexplained and the possibility of equivalent (non) holonomic formulations of gravitytheories is analyzed.

Section 3 is devoted to nonholonomic deformations of manifolds and vectorbundles. The basic constructions in the geometry of (generalized) Lagrange andFinsler spaces are reviewed. A general ansatz for constructing exact solutions, witheffective Lagrange and Finsler structures, in Einstein and string gravity, is analyzed.

In Sec. 4, the Finsler–Lagrange geometry is formulated as a variant of almostHermitian and/or Kaher geometry. We show how the Einstein gravity can beequivalently reformulated in terms of almost Hermitian geometry with preferredframe structure.


478 S. I. Vacaru

Section 5 is focused on explicit examples of exact solutions in Einstein and stringgravity when (generalized) Finsler–Lagrange structures are modeled on (pseudo)Riemannian and Riemann–Cartan spaces. We analyze some classes of Einsteinmetrics which can be deformed into new exact solutions characterized addition-ally by Lagrange–Finsler configurations. For string gravity, there are constructedexplicit examples of locally anisotropic configurations describing gravitational soli-tonic pp-waves and their effective Lagrange spaces. We also analyze some exactsolutions for Finsler-solitonic pp-waves on Schwarzschild spaces.

Conclusions and further perspectives of Finsler geometry and new geometricmethods for modern gravity theories are considered in Sec. 6.

Finally, we should note that our list of references is minimal, trying to con-centrate on reviews and monographs rather than on original articles. More com-plete reference lists are presented in the books [38, 39, 62, 76, 77]. Various guidesfor learning, both for experts and beginners on geometric methods and furtherapplications in standard and nonstandard physics, with references, are containedin [10, 38, 39, 53, 77].

2. Nonholonomic Einstein Gravity and Finsler–Lagrange Spaces

In this section we present in a unified form the Riemann–Cartan and Finsler–Lagrange geometry. The reader is supposed to be familiar with well-known geo-metrical approaches to gravity theories [21, 40, 56, 83] but may not know the basicconcepts on Finsler geometry and nonholonomic manifolds. The constructions forlocally anisotropic spaces will be derived by special parametrizations of the frame,metric and connection structures on usual manifolds, or vector bundle spaces, aswe proved in details in [64, 77].

2.1. Metric-affine, Riemann–Cartan and Einstein manifolds

Let V be a necessary smooth class manifold of dimension dimV = n + m, whenn ≥ 2 and m ≥ 1, enabled with metric, g = gαβe

α ⊗ eβ, and linear connection,D = Γαβγ, structures. The coefficients of g and D can be computed with respectto any local frame, eα, and co–frame, eβ , bases, for which eαeβ = δβα, where denotes the interior (scalar) product defined by g and δβα is the Kronecker symbol.A local system of coordinates on V is denoted uα = (xi, ya), or (in brief) u =(x, y), where indices run correspondingly the values: i, j, k . . . = 1, 2, . . . , n anda, b, c, . . . = n + 1, n + 2, . . . n + m for any splitting α = (i, a), β = (j, b), . . . . Weshall also use primed, underlined, or other type indices: for instance, eα′ = (ei′ , ea′)and eβ

′= (ej

′, eb

′), for a different set of local (co) bases, or eα = eα = ∂α = ∂/∂uα,

ei = ei = ∂i = ∂/∂xi and ea = ea = ∂a = ∂/∂ya if we want to emphasize that thecoefficients of geometric objects (tensors, connections, . . . ) are defined with respectto a local coordinate basis. For simplicity, we shall omit underlining or priming ofindices and symbols if that will not result in ambiguities. The Einstein’s summationrule on repeating “up-low” indices will be applied if the contrary will not be stated.



Frame transforms of a local basis eα and its dual basis eβ are parametrizedin the form

eα = A α′α (u)eα′ and eβ = Aββ′(u)eβ

′, (1)

where the matrix Aββ′ is inverse to A α′α . In general, local bases are nonholonomic

(equivalently, anholonomic, or nonintegrable) and satisfy certain anholonomyconditions

eαeβ − eβeα = W γαβeγ (2)

with nontrivial anholonomy coefficients W γαβ(u). We consider the holonomic

frames to be defined by W γαβ = 0, which holds, for instance, if we fix a local

coordinate basis.Let us denote the covariant derivative along a vector field X = Xαeα as

DX = XD. One defines three fundamental geometric objects on manifold V :nonmetricity field,

QX DXg, (3)

torsion,

T (X,Y ) DXY −DYX − [X,Y ], (4)

and curvature,

R(X,Y )Z DXDY Z −DYDXZ −D[X,Y ]Z, (5)

where the symbol “” states “by definition” and [X,Y ] XY −Y X . With respectto fixed local bases eα and eβ, the coefficients Q = Qαβγ = Dαgβγ, T = Tαβγand R = Rαβγτ can be computed by introducing X → eα, Y → eβ , Z → eγ intorespective formulas (3), (4) and (5).

In gravity theories, one uses three other important geometric objects: the Riccitensor, Ric(D) = R βγ Rαβγα, the scalar curvature, R gαβRαβ (gαβ beingthe inverse matrix to gαβ), and the Einstein tensor, E = Eαβ Rαβ − 1

2gαβR.A manifold maV is a metric-affine space if it is provided with arbitrary two

independent metric g and linear connectionD structures and characterized by threenontrivial fundamental geometric objects Q, T and R.

If the metricity condition, Q = 0, is satisfied for a given couple g and D, sucha manifold RCV is called a Riemann–Cartan space with nontrivial torsion Tof D.

A Riemann space RV is provided with a metric structure g which defines aunique Levi–Civita connection D = ∇, which is both metric compatible, Q =∇g = 0, and torsionless, T = 0. Such a space is pseudo- (semi-) Riemannian iflocally the metric has any mixed signature (±1,±1, . . . ,±1).c In brief, we shall

cMathematicians usually use the term semi-Riemannian but physicists are more familiar withpseudo-Riemannian; we shall apply both terms on convenience.


480 S. I. Vacaru

call all such spaces to be Riemannian (with necessary signature) and denote themain geometric objects in the form R = R

αβγτ, Ric(D) = R βγ, R and

E = Eαβ.The Einstein gravity theory is constructed canonically for dimR V = 4

and Minkowski signature, for instance, (−1,+1,+1,+1). Various generalizationsin modern string and/or gauge gravity consider Riemann, Riemann–Cartanand metric–affine spaces of higher dimensions.

The Einstein equations are postulated in the form

E(D) Ric(D) − 12g Sc(D) = Υ, (6)

where the source Υ contains contributions of matter fields and corrections from,for instance, string/brane theories of gravity. In a physical model, Eq. (6) has tobe completed with equations for the matter fields and torsion (for instance, in theEinstein–Cartan theory [22], one considers algebraic equations for the torsionand its source). It should be noted here that because of possible nonholonomic struc-tures on a manifold V (we shall call such spaces to be locally anisotropic), see nextsection, the tensor Ric(D) is not symmetric and D[E(D)] = 0. This imposes a moresophisticate form of conservation laws on spaces with generic “local anisotropy”,see discussion in [77] (a similar situation arises in Lagrange mechanics [29–32, 38]when nonholonomic constraints modify the definition of conservation laws).

For general relativity, dimV = 4 and D = ∇, the field equations can bewritten in the well-known component form

Eαβ = R βγ −12 R = Υαβ (7)

when ∇(Eαβ) = ∇(Υαβ) = 0. The coefficients in Eq. (7) are defined with respectto arbitrary nonholomomic frame (1).

2.2. Nonholonomic manifolds and adapted frame structures

A nonholonomic manifold (M,D) is a manifold M of necessary smooth classenabled with a nonholonomic distribution D, see details in [11,77]. Let us considera (n+m)-dimensional manifold V, with n ≥ 2 and m ≥ 1 (for a number of physicalapplications, it will be considered to model a physical or geometric space). In aparticular case, V = TM , with n = m (i.e. a tangent bundle), or V = E = (E,M),dimM = n, is a vector bundle on M , with total space E (we shall use such spacesfor traditional definitions of Finsler and Lagrange spaces [8, 10, 33, 37, 38, 53, 54]).In a general case, a manifold V is provided with a local fibred structure into con-ventional “horizontal” and “vertical” directions defined by a nonholonomic (nonin-tegrable) distribution with associated nonlinear connection (equivalently, nonholo-nomic frame) structure. Such nonholonomic manifolds will be used for modelinglocally anisotropic structures in Einstein gravity and generalizations [61,64,77–79].



2.2.1. Nonlinear connections and N-adapted frames

We denote by π : TV → TM the differential of a map π : V → M defined byfiber preserving morphisms of the tangent bundles TV and TM . The kernel of π

is just the vertical subspace vV with a related inclusion mapping i : vV → TV. Forsimplicity, in this work we restrict our considerations for a fibered manifold V →M

with constant rank π. In such cases, we can define connections and metrics on Vin usual form, but with the aim to “mimic” Finsler and Lagrange like structures(not on usual tangent bundles but on such nonholonomic manifolds) we shall alsoconsider metrics, tensors and connections adapted to the fibered structure as itwas elaborated in Finsler geometry (see below the concept of distinguished metric,Subsec. 2.2.2 and distinguished connection, Subsec. 2.2.3).

A nonlinear connection (N-connection) N on a manifold V is defined bythe splitting on the left of an exact sequence

0 → vV i→ TV → TV/vV → 0,

i.e. by a morphism of submanifolds N : TV → vV such that N i is the unity invV.d

Locally, a N-connection is defined by its coefficients Nai (u),

N = Nai (u)dxi ⊗ ∂

∂ya. (8)

In an equivalent form, we can say that any N-connection is defined by a Whitneysum of conventional horizontal (h) space, (hV), and vertical (v) space, (vV),

TV = hV ⊕ vV. (9)

The sum (9) states on TV a nonholonomic (equivalently, anholonomic, or noninte-grable) distribution of h- and v-space. The well known class of linear connectionsconsists on a particular subclass with the coefficients being linear on ya, i.e.

Nai (u) = Γabj(x)y

b. (10)

The geometric objects on V can be defined in a form adapted to a N-connectionstructure, following decompositions which are invariant under parallel transportspreserving the splitting (9). In this case, we call them to be distinguished (by theN-connection structure), i.e. d-objects. For instance, a vector field X ∈ TV isexpressed

X = (hX, vX), or X = Xαeα = X iei +Xaea,

dThere is a proof (see, for instance, [38], Theorem 1.2, p. 21) that for a vector bundle over aparacompact manifold there exist N-connections. In this work, we restrict our considerations onlyfor fibered manifolds admitting N-connection and related N-adapted frame structures (see the endof this section).


482 S. I. Vacaru

where hX = X iei and vX = Xaea state, respectively, the adapted to the N-connection structure horizontal (h) and vertical (v) components of the vector. Inbrief, X is called a distinguished vector, d-vector.e In a similar fashion, the geo-metric objects on V like tensors, spinors, connections,. . . are called respectivelyd-tensors, d-spinors, d-connections if they are adapted to the N-connectionsplitting (9).

The N-connection curvature is defined as the Neijenhuis tensor

Ω(X,Y) [vX, vY ] + v[X,Y] − v[vX,Y] − v[X,vY ]. (11)

In local form, we have for (11) Ω = 12Ωaij d

i ∧ dj ⊗ ∂a, with coefficients

Ωaij =∂Na

i

∂xj−∂Na

j

∂xi+N b

i

∂Naj

∂yb−N b

j

∂Nai

∂yb. (12)

Any N-connection N may be characterized by an associated frame (vielbein)structure eν = (ei, ea), where

ei =∂

∂xi−Na

i (u)∂

∂yaand ea =

∂

∂ya, (13)

and the dual frame (coframe) structure eµ = (ei, ea), where

ei = dxi and ea = dya +Nai (u)dxi, (14)

see formulas (1). These vielbeins are called respectively N-adapted frames andcoframes. In order to preserve a relation with the previous denotations [64, 77],we emphasize that eν = (ei, ea) and eµ = (ei, ea) are correspondingly the former“N-elongated” partial derivatives δν = δ/∂uν = (δi, ∂a) and N-elongated differen-tials δµ = δuµ = (di, δa). This emphasizes that the operators (13) and (14) definecertain “N-elongated” partial derivatives and differentials which are more conve-nient for tensor and integral calculations on such nonholonomic manifolds. Thevielbeins (14) satisfy the nonholonomy relations

[eα, eβ ] = eαeβ − eβeα = W γαβeγ (15)

with (antisymmetric) nontrivial anholonomy coefficients W bia = ∂aN

bi and W a

ji =Ωaij defining a proper parametrization (for a n+m splitting by a N-connection Na

i )of (14).

2.2.2. N-anholonomic manifolds and d-metrics

For simplicity, we shall work with a particular class of nonholonomic manifolds:A manifold V is N-anholonomic if its tangent space TV is enabled with aN-connection structure (9).f

eWe shall always use “boldface” symbols if it would be necessary to emphasize that certain spacesand/or geometrical objects are provided/adapted to a N–connection structure, or with the coeffi-cients computed with respect to N–adapted frames.f In a similar manner, we can consider different types of (super) spaces and low energy stringlimits [59, 60, 62, 73], Riemann or Riemann–Cartan manifolds [77], noncommutative bundles, or



A distinguished metric (in brief, d-metric) on a N-anholonomic manifoldV is a usual second rank metric tensor g which with respect to a N–adapted basis(14) can be written in the form

g = gij(x, y) ei ⊗ ej + hab(x, y) ea ⊗ eb (16)

defining a N-adapted decomposition g =hg⊕Nvg = [hg, vg].A metric structure g on a N-anholonomic manifold V is a symmetric covariant

second rank tensor field which is not degenerated and of constant signature in anypoint u ∈ V. Any metric on V, with respect to a local coordinate basis duα =(dxi, dya), can be parametrized in the form

g = gαβ

(u)duα ⊗ duβ (17)

where

gαβ

=

[gij +Na

i Nbj hab Ne

j hae

Nei hbe hab

]. (18)

Such a metric (18) is generic off-diagonal, i.e. it cannot be diagonalized by coordi-nate transforms if Na

i (u) are any general functions.In general, a metric structure is not adapted to a N-connection structure, but

we can transform it into a d-metric

g = hg(hX, hY ) + vg(vX, vY ) (19)

adapted to a N-connection structure defined by coefficients Nai . We introduce deno-

tations hg(hX, hY ) = hg(hX, hY ) and vg(vX , vY ) = vg(vX, vY ) and try to finda N-connection when

g(hX, vY ) = 0 (20)

for any d-vectors X,Y. In local form, for hX → ei and vY → ea, Eq. (20) is analgebraic equation for the N-connection coefficients Na

i ,

g(ei, ea) = 0, equivalently, gia−N b

i hab = 0, (21)

where gia

g(∂/∂xi, ∂/∂ya), which allows us to define in a unique form the coef-ficients N b

i = habgia

where hab is inverse to hab. We can write the metric g withansatz (18) in equivalent form, as a d-metric (16) adapted to a N-connection struc-ture, if we define gij g(ei, ej) and hab g(ea, eb) and consider the vielbeins eαand eα to be respectively of type (13) and (14).

superbundles and gauge models [17, 63, 64, 74, 75], Clifford–Dirac spinor bundles and algebroids[58, 61, 65, 76, 78, 79], Lagrange–Fedosov manifolds [19]. . . provided with nonholonomc (super)distributions (9) and preferred systems of reference (supervielbeins).


484 S. I. Vacaru

A metric g (17) can be equivalently transformed into a d-metric (16) by per-forming a frame (vielbein) transform

eα = e αα ∂α and eβ = eββdu

β , (22)

with coefficients

e αα (u) =

[eii (u) N b

i (u)e ab (u)

0 eaa (u)

], (23)

eββ(u) =

[eii(u) −N b

k(u)eki (u)

0 eaa(u)

], (24)

being linear on Nai .

It should be noted here that parametrizations of metrics of type (18) havebeen introduced in Kaluza–Klein gravity [46] for the case of linear connections(10) and compactified extra dimensions ya. For the five (or higher) dimensions, thecoefficients Γabi(x) were considered as Abelian or non-Abelian gauge fields. In ourapproach, the coefficientsN b

i (x, y) are general ones, not obligatory linearized and/orcompactified on ya. For some models of Finsler gravity, the values Na

i were treatedas certain generalized nonlinear gauge fields (see Appendix to [37]), or as certainobjects defining (semi) spray configurations in generalized Finsler and Lagrangegravity [3, 37, 38].

On N-anholonomic manifolds, we can say that the coordinates xi are holonomicand the coordinates ya are nonholonomic (on N-anholonomic vector bundles, suchcoordinates are called respectively to be the horizontal and vertical ones). We con-clude that a N-anholonomic manifold V provided with a metric structure g (17)(equivalently, with a d-metric (16)) is a usual manifold (in particular, a pseudo–Riemannian one) with a prescribed nonholonomic n+m splitting into conventional“horizontal” and “vertical” subspaces (9) induced by the “off-diagonal” termsN b

i (u)and the corresponding preferred nonholonomic frame structure (15).

2.2.3. d-torsions and d-curvatures

From the general class of linear connections which can be defined on a manifold V ,and any of its N-anholonomic versions V, we distinguish those which are adaptedto a N-connection structure N.

A distinguished connection (d-connection) D on a N-anholonomic mani-fold V is a linear connection conserving under parallelism the Whitney sum (9). Forany d-vector X, there is a decomposition of D into h- and v-covariant derivatives,

DX XD = hXD+ vXD =DhX +DvX = hDX + vDX . (25)



The symbol “” in (25) denotes the interior product defined by a metric (17) (equiv-alently, by a d-metric (16)). The N-adapted components Γαβγ of a d-connectionDα = (eαD) are defined by the equations

Dαeβ = Γγαβeγ , or Γγαβ (u) = (Dαeβ)eγ . (26)

The N-adapted splitting into h- and v-covariant derivatives is stated by

hD = Dk = (Lijk, Labk ), and vD = Dc =

(Cijc, C

abc

),

where Lijk = (Dkej)ei, Labk = (Dkeb)ea, Cijc = (Dcej)ei, Cabc = (Dceb)ea. Thecomponents Γγαβ = (Lijk, L

abk, C

ijc, C

abc) completely define a d-connection D on a

N-anholonomic manifold V. We shall write conventionally that D =(hD, vD), orDα = (Di, Da), with hD = (Lijk, L

abk) and vD = (Cijc, C

abc), see (26).

The torsion and curvature of a d-connection D = (hD, vD), d-torsions andd-curvatures, are defined similarly to formulas (4) and (5) with further h- andv-decompositions. The simplest way to perform computations with d-connectionsis to use N-adapted differential forms like

Γαβ = Γαβγeγ (27)

with the coefficients defined with respect to (14) and (13). For instance, torsion canbe computed in the form

T α Deα = deα + Γαβ ∧ eβ. (28)

Locally it is characterized by (N-adapted) d-torsion coefficients

T ijk = Lijk − Likj , T ija = −T iaj = Cija, T aji = Ωaji,

T abi = −T aib =∂Na

i

∂yb− Labi, T abc = Cabc − Cacb.

(29)

By a straightforward d-form calculus, we can compute the N-adapted componentsR = Rα

βγδ of the curvature

Rαβ DΓαβ = dΓαβ − Γγβ ∧ Γαγ = Rα

βγδeγ ∧ eδ, (30)

of a d-connection D,

Rihjk = ekLihj − ejL

ihk + LmhjL

imk − LmhkL

imj − CihaΩ

akj ,

Rabjk = ekLabj − ejL

abk + LcbjL

ack − LcbkL

acj − CabcΩ

ckj ,

Rijka = eaLijk −DkC

ija + CijbT

bka,

Rcbka = eaLcbk −DkC

cba + CcbdT

cka,

Rijbc = ecCijb − ebC

ijc + ChjbC

ihc − ChjcC

ihb,

Rabcd = edCabc − ecC

abd + CebcC

aed − CebdC

aec.

(31)

Contracting respectively the components of (31), one proves that the Ricci ten-sor Rαβ Rτ

αβτ is characterized by h- v-components

Rij Rkijk, Ria −Rkika, Rai Rbaib, Rab Rcabc. (32)


486 S. I. Vacaru

It should be noted that this tensor is not symmetric for arbitrary d-connections D,i.e. Rαβ = Rβα. The scalar curvature of a d-connection is

sR gαβRαβ = gijRij + habRab, (33)

defined by a sum the h- and v-components of (32) and d-metric (16).The Einstein d-tensor is defined and computed similarly to (7), but for d-

connections,

Eαβ = Rαβ − 12gαβsR. (34)

This d-tensor defines an alternative to Eαβ (nonholonomic) Einstein configurationif its d-connection is defined in a unique form for an off-diagonal metric (18).

2.2.4. Some classes of distinguished or non-adapted linear connections

From the class of arbitrary d-connections D on V, one distinguishes those whichare metric compatible (metrical d-connections) satisfying the condition

Dg = 0 (35)

including all h- and v-projections Djgkl = 0, Dagkl = 0, Djhab = 0, Dahbc = 0.Different approaches to Finsler–Lagrange geometry modeled on TM (or on the dualtangent bundle T∗M, in the case of Cartan–Hamilton geometry) were elaboratedfor different d-metric structures which are metric compatible [15, 35–39, 60, 62, 76]or not metric compatible [8].

For any d-metric g = [hg, vg] on a N-anholonomic manifold V, there is a uniquemetric canonical d-connection D satisfying the conditions Dg =0 and with vanish-ing h(hh)-torsion, v(vv)-torsion, i.e. hT (hX, hY ) = 0 and vT (vX, vY ) = 0. Bystraightforward calculations, we can verify that Γγαβ = (Lijk, L

abk, C

ijc, C

abc), when

Lijk =12gir(ekgjr + ejgkr − ergjk),

Labk = eb(Nak ) +

12hac(ekhbc − hdc ebN

dk − hdb ecN

dk ),

Cijc =12gikecgjk, C

abc =

12had(echbd + echcd − edhbc)

(36)

result in T ijk = 0 and T abc = 0 but T ija, Taji and T abi are not zero, see formulas

(29) written for this canonical d-connection.For any metric structure g on a manifold V, there is a unique metric compatible

and torsionless Levi–Civita connection = Γαβγ for which T = 0 and g =0. This is not a d-connection because it does not preserve under parallelism theN-connection splitting (9) (it is not adapted to the N-connection structure). Letus parametrize its coefficients in the form

Γαβγ =(

Lijk, L

ajk, L

ibk, L

abk, C

ijb, C

ajb, C

ibc, C

abc

),



where ek(ej) = L

ijkei + L

ajkea, ek

(eb) = Libkei + L

abkea,eb

(ej) = Cijbei +

Cajbea, ec(eb) = C

ibcei+ C

abcea. A straightforward calculusg shows that the coef-

ficients of the Levi–Civita connection are

Lijk = Lijk, L

ajk = −Cijbgikhab −

12Ωajk,

Libk =

12Ωcjkhcbg

ji − 12(δijδ

hk − gjkg

ih)Cjhb,

Labk = Labk +

12(δac δ

bd + hcdh

ab)[Lcbk − eb(N ck)],

Cikb = Cikb +

12Ωajkhcbg

ji +12(δijδ

hk − gjkg

ih)Cjhb,

Cajb = −1

2(δac δ

db − hcbh

ad)[Lcdj − ed(N cj )], C

abc = Cabc,

Ciab = −g

ij

2[Lcaj − ea(N c

j )]hcb +

[Lcbj − eb(N c

j )]hca

,

(37)

where Ωajk are computed as in formula (12). For certain considerations, it is conve-nient to express

Γγαβ = Γγαβ + Z

γαβ (38)

where the explicit components of distorsion tensor Zγαβ can be defined by com-

paring the formulas (37) and (36). It should be emphasized that all components ofΓγαβ , Γ

γαβ andZ

γαβ are uniquely defined by the coefficients of d-metric (16) and

N-connection (8), or equivalently by the coefficients of the corresponding genericoff-diagonal metric (18).

2.3. On equivalent (non)holonomic formulations of gravity

theories

A N-anholonomic Riemann-Cartan manifold RCV is defined by a d-metric gand a metric d-connection D structures. We can say that a spaceRV is a canon-ical N-anholonomic Riemann manifold if its d-connection structure is canonical,i.e. D= D. The d-metric structure g onRCV is of type (16) and satisfies themetricity conditions (35). With respect to a local coordinate basis, the metric gis parametrized by a generic off-diagonal metric ansatz (18). For a particular case,we can treat the torsion T as a nonholonomic frame effect induced by a noninte-grable N-splitting. We conclude that a manifold RV is enabled with a nontrivialtorsion (29) (uniquely defined by the coefficients of N-connection (8), and d-metric(16) and canonical d-connection (36) structures). Nevertheless, such manifolds canbe described alternatively, equivalently, as a usual (holonomic) Riemann manifold

gSuch results were originally considered by Miron and Anastasiei for vector bundles providedwith N-connection and metric structures, see [38]. Similar proofs hold true for any nonholonomicmanifold provided with a prescribed N-connection structure [77].


488 S. I. Vacaru

with the usual Levi–Civita for the metric (17) with coefficients (18). We do notdistinguish the existing nonholonomic structure for such geometric constructions.

Having prescribed a nonholonomic n+m splitting on a manifold V , we can definetwo canonical linear connections ∇ and D. Correspondingly, these connections arecharacterized by two curvature tensors, R

αβγδ(∇) (computed by introducing Γαβγ

into (27) and (30)) and Rαβγδ(D) (with the N-adapted coefficients computed follow-

ing formulas (31)). Contracting indices, we can commute the Ricci tensor Ric(∇)and the Ricci d-tensor Ric(D) following formulas (32), correspondingly written for∇ and D. Finally, using the inverse d-tensor gαβ for both cases, we compute the cor-responding scalar curvatures sR(∇) and sR(D), see formulas (33) by contracting,respectively, with the Ricci tensor and Ricci d-tensor.

The standard formulation of the Einstein gravity is for the connection ∇,when the field equations are written in the form (7). But it can be equivalentlyreformulated by using the canonical d-connection, or other connections uniquelydefined by the metric structure. If a metric (18) g

αβis a solution of the Einstein

equationsEαβ = Υαβ , having prescribed a (n + m)-decomposition, we can definealgebraically the coefficients of a N-connection, Na

i , N-adapted frames eα (13) andeβ (14), and d-metric gαβ = [gij , hab] (16). The next steps are to compute Γγαβ ,

following formulas (36), and then using (31), (32) and (33) for D, to define Eαβ

(34). The Einstein equations with matter sources, written in equivalent form byusing the canonical d-connection, are

Eαβ = Υαβ + ZΥαβ , (39)

where the effective source ZΥαβ is just the deformation tensor of the Einsteintensor computed by introducing deformation (38) into the left part of (7); alldecompositions being performed with respect to the N-adapted co-frame (14), whenEαβ = Eαβ−ZΥαβ . For certain matter field/string gravity configurations, the solu-tions of (39) also solve Eq. (7). Nevertheless, because of generic nonlinear characterof gravity and gravity-matter field interactions and functions defining nonholonomicdistributions, one could be certain special conditions when even vacuum configura-tions contain a different physical information if to compare with usual holonomicones. We analyze some examples:

In our works [64, 65, 77], we investigated a series of exact solutions definingN-anholonomic Einstein spaces related to generic off-diagonal solutions in generalrelativity by such nonholonomic constraints when Ric(D) = Ric(∇), even D = ∇.h

In this case, for instance, the solutions of the Einstein equations with cosmologicalconstant λ,

Rαβ = λgαβ (40)

hOne should emphasize here that different types of connections on N-anholonomic manifolds havedifferent coordinate and frame transform properties. It is possible, for instance, to get equalitiesof coefficients for some systems of coordinates even the connections are very different. The trans-formation laws of tensors and d-tensors are also different if some objects are adapted and othersare not adapted to a prescribed N-connection structure.



can be transformed into metrics for usual Einstein spaces with Levi–Civitaconnection ∇. The idea is that for certain general metric ansatz, see Subsec. 3.2,Eq. (40) can be integrated in general form just for the connection D but not for ∇.The nontrivial torsion components

T ija = −T iaj = Cija, T aji = T aij = Ωaji, T abi = −T aib =∂Na

i

∂yb− Labi, (41)

see (29), for some configurations, may be associated with an absolute antisymmetricH-fields in string gravity [18, 51], but nonholonomically transformed to N-adaptedbases, see details in [64, 77].

For more restricted configurations, we can search solutions with metric ansatzdefining Einstein foliated spaces, when

Ωcjk = 0, Lcbk = eb(N ck), Cijb = 0, (42)

and the d-torsion components (41) vanish, but the N-adapted frame structure has,in general, nontrivial anholonomy coefficients, see (15). One present a special inter-est a less constrained configurations with T cjk = Ωcjk = 0 when Ric(D) = Ric(∇)and T ijk = T abc = 0, for certain general ansatz T ija = 0 and T abi = 0, butRα

βγδ = Rαβγδ. In such cases, we constrain the integral varieties of Eq. (40) in

such a manner that we generate integrable or nonintegrable distributions on a usualEinstein space defined by ∇. This is possible because if the conditions (42) are sat-isfied, the deformation tensor Z

γαβ = 0. For λ = 0, if n+m = 4, for corresponding

signature, we get foliated vacuum configurations in general relativity.For N-anholonomic manifolds Vn+n of odd dimensions, when m = n, and if

gij = hij (we identify correspondingly, the h- and v-indices), we can consider acanonical d-connection D = (hD, vD) with the nontrivial coefficients with respectto eν and eµ parametrized respectively Γαβγ = (Lijk = Labk, C

ijc = Cabc),

i for

Lijk =12gih(ekgjh + ejgkh − ehgjk), Cabc =

12gae(ebgec + ecgeb − eegbc), (43)

defining the generalized Christoffel symbols. Such nonholonomic configurations canbe used for modeling generalized Finsler–Lagrange, and particular cases, definedin [37, 38] for Vn+n = TM, see Subsec. 3.1 below. There are only three classes ofd-curvatures for the d-connection (43),

Rihjk = ekLihj − ejLihk + LmhjLimk − LmhkL

imj − CihaΩ

akj , (44)

P ijka = eaLijk − DkC

ija, S

abcd = edC

abc − ecC

abd + CebcC

aed − CebdC

aec,

where all indices a, b, . . . , i, j, . . . run the same values and, for instance, Cebc →Cijk, . . . . Such locally anisotropic configurations are not integrable if Ωakj = 0,even the d-torsion components T ijk = 0 and T abc = 0. We note that for geometricmodels on Vn+n, or on TM, with gij = hij , one writes, in brief, Γαβγ = (Lijk, C

abc),

iThe equalities of indices “i = a” are considered in the form “i = 1 = a = n + 1, i = 2 = a =n + 2, . . . i = n = a = n + n”.


490 S. I. Vacaru

or, for more general d-connections, Γαβγ = (Lijk, Cabc), see Subsec. 3.1 below, on

Lagrange and Finsler spaces.

3. Nonholonomic Deformations of Manifolds and Vector Bundles

This section will deal mostly with nonholonomic distributions on manifolds and vec-tor/tangent bundles and their nonholonomic deformations modeling, on Riemannand Riemann–Cartan manifolds, different types of generalized Finsler–Lagrangegeometries.

3.1. Finsler–Lagrange spaces and generalizations

The notion of Lagrange space was introduced by Kern [27] and elaborated indetails by Miron’s school, see [34–39], as a natural extension of Finsler geometry[10,15,33,53] (see also [60,62,64,73,77], on Lagrange–Finsler super/noncommutativegeometry). Originally, such geometries were constructed on tangent bundles, butthey also can be modeled on N-anholonomic manifolds, for instance, as mod-els for certain gravitational interactions with prescribed nonholonomic constraintsdeformed symmetries.

3.1.1. Lagrange spaces

A differentiable Lagrangian L(x, y), i.e. a fundamental Lagrange function, isdefined by a map L : (x, y) ∈ TM → L(x, y) ∈ R of class C∞ on TM = TM\0and continuous on the null section 0 : M → TM of π. A regular Lagrangian hasnon-degenerate Hessian

Lgij(x, y) =12∂2L(x, y)∂yi∂yj

, (45)

when rank|gij | = n and Lgij is the inverse matrix. A Lagrange space is a pairLn = [M,L(x, y)] with Lgij being of fixed signature over V = TM .

One holds the result: The Euler–Lagrange equations ddτ ( ∂L∂yi ) − ∂L

∂xi = 0,

where yi = dxi

dτ for xi(τ) depending on parameter τ , are equivalent to the “nonlin-ear” geodesic equations d2xa

dτ2 + 2Ga(xk, dxb

dτ ) = 0 defining paths of a canonicalsemispray S = yi ∂

∂xi − 2Ga(x, y) ∂∂ya , where 2Gi(x, y) = 1

2Lgij( ∂2L

∂yi∂xk yk − ∂L

∂xi ).

There exists on V TM a canonical N-connection

LNaj =

∂Ga(x, y)∂yj

(46)

defined by the fundamental Lagrange function L(x, y), which prescribes nonholo-nomic frame structures of type (13) and (14), Leν = (Lei, ea) and Leµ = (ei, Lea).One defines the canonical metric structure

Lg = Lgij(x, y) ei ⊗ ej + Lgij(x, y)Lei ⊗ Lej (47)

constructed as a Sasaki type lift from M for Lgij(x, y), see details in [37, 38, 86].



There is a unique canonical d-connectionLD = (hLD, vLD) with the coeffi-cients LΓαβγ = (LLijk,

LCabc) computed by formulas (43) for the d-metric (47)with respect to Leν and Leµ. All such geometric objects, including the corre-sponding to LΓαβγ ,

Lg and LNaj d-curvatures LRα

βγδ =(LRihjk,LP ijka,

LSabcd), see(44), are completely defined by a Lagrange fundamental function L(x, y) for a non-degerate Lgij .

We conclude that any regular Lagrange mechanics can be geometrized as a non-holonomic Riemann manifold LV equipped with the canonical N-connection LNa

j

(46). This geometrization was performed in such a way that the N-connection isinduced canonically by the semispray configurations subjected the condition thatthe generalized nonlinear geodesic equations are equivalent to the Euler–Lagrangeequations for L. Such mechanical models and semispray configurations can be usedfor a study of certain classes of nonholonomic effective analogous of gravitationalinteractions. The approach can be extended for more general classes of effective met-rics, then those parametrized by (47), see next sections. After Kern and Miron andAnastasiei works, it was elaborated the so-called “analogous gravity” approach [9]with similar ideas modeling related to continuous mechanics, condensed media. . . .It should be noted here, that the constructions for higher order generalized Lagrangeand Hamilton spaces [34–36,39] provided a comprehensive geometric formalism foranalogous models in gravity, geometric mechanics, continuous media, nonhomoge-neous optics, etc.

3.1.2. Finsler spaces

Following the ideas of the Romanian school on Finsler–Lagrange geometry andgeneralizations, any Finsler space defined by a fundamental Finsler functionF (x, y), being homogeneous of type F (x, λy) = |λ| F (x, y), for nonzero λ ∈ R, maybe considered as a particular case of Lagrange space when L = F 2 (on differentrigorous mathematical definitions of Finsler spaces, see [8, 10, 33, 37, 38, 53]; in ourapproach with applications to physics, we shall not constrain ourself with specialsignatures, smooth class conditions and special types of connections). Historically,the bulk of mathematicians worked in an inverse direction, by generalizing the con-structions from the Cartan’s approach to Finsler geometry in order to include intoconsideration regular Lagrange mechanical systems, or to define Finsler geometrieswith another type of nonlinear and linear connection structures. The Finsler geome-try, in terms of the normal canonical d-connection (43), derived for respective F gijand FNa

j , can be modeled as for the case of Lagrange spaces considered in theprevious section: we have to change formally all labels L → F and take into con-sideration possible conditions of homogeneity (or TM , see the monographs [37,38]).

For generalized Finsler spaces, a N-connection can be stated by a general setof coefficients Na

j subjected to certain nonholonomy conditions. Of course, workingwith homogeneous functions on a manifold V n+n, we can model a Finsler geometryboth on holonomic and nonholonomic Riemannian manifolds, or on certain types of


492 S. I. Vacaru

Riemann–Cartan manifolds enabled with preferred frame structures Feν = (F ei, ea)and Feµ = (ei, Fea). Below, in the Subsec. 3.2, we shall discuss how certain typeof Finsler configurations can be derived as exact solutions in Einstein gravity. Suchconstructions allow us to argue that Finsler geometry is also very important instandard physics and that it was a big confusion to treat it only as a “sophisti-cated” generalization of Riemann geometry, on tangent bundles, with not muchperspectives for modern physics.

In a number of works (see monographs [8, 37, 38]), it is emphasized that thefirst example of Finsler metric was considered in the famous inauguration thesisof Riemann [52], long time before Finsler [20]. Perhaps, this is a reason, for manyauthors, to use the term Riemann–Finsler geometry. Nevertheless, we would like toemphasize that a Finsler space is not completely defined only by a metric structureof type

F gij =12

∂2F

∂yi∂yj(48)

originally considered on the vertical fibers of a tangent bundle. There are necessaryadditional conventions about metrics on a total Finsler space, N-connections andlinear connections. This is the source for different approaches, definitions, construc-tions and ambiguities related to Finsler spaces and applications. Roughly speak-ing, different famous mathematicians, and their schools, elaborated their versionsof Finsler geometries following some special purposes in geometry, mechanics andphysics.

The first complete model of Finsler geometry exists due to Cartan [15] whoin the 20–30th years of previous century elaborated the concepts of vector bun-dles, Rieman–Cartan spaces with torsion, moving frames, developed the theory ofspinors, Pfaff forms . . . and (in coordinate form) operated with nonlinear connec-tion coefficients. The Cartan’s constructions were performed with metric compatiblelinear connections which is very important for applications to standard models inphysics.

Later, there were proposed different models of Finsler spaces with metricnot compatible linear connections. The most notable connections were those byBerwald, Chern (re-discovered by Rund), Shimada and others (see details, discus-sions and bibliography in monographs [8,37,38,53]). For d-connections of type (43),there are distinguished three cases of metric compatibility (compare with h- andv-projections of formula (35)): A Finsler connection FDα = (FDk,

FDa) is calledh-metric if FDF

i gij = 0; it is called v-metric if FDaF gij = 0 and it is metrical if

both conditions are satisfied.Here, we note three of the most important Finsler d-connections having their

special geometric and (possible) physical merits:

1. The canonical Finsler connection F D is defined by formulas (43), but forF gij , i.e. as F Γαβγ = (F Lijk,

F Cabc). This d-connection is metrical. For a spe-cial class of N-connections CNa

j (xk, yb) = ykCLikj , we get the famous Cartan



connection for Finsler spaces, CΓαβγ = (CLijk,CCabc), with

CLijk =12F gih(CekF gjh + CejF gkh − CehF gjk),

CCabc =12F gae(ebF gec + ec

F geb − eeF gbc),

(49)

where Cek = ∂∂xk − CNa

j∂∂ya and eb = ∂

∂yb , which can be defined in a uniqueaxiomatic form [33]. Such canonical and Cartan–Finsler connections, being met-ric compatible, for nonholonomic geometric models with local anisotropy onRiemann or Riemann–Cartan manifolds, are more suitable with the paradigmof modern standard physics.

2. The Berwald connection BD was introduced in the form BΓαβγ = (∂CNb

j

∂ya , 0)[13]. This d-connection is defined completely by the N-connection structure butit is not metric compatible, both not h-metric and not v-metric.

3. The Chern connection ChD was considered as a minimal Finsler extension ofthe Levi–Civita connection, ChΓαβγ = (CLijk, 0), with CLijk defined as in (49),preserving the torsionless condition, being h-metric but not v-metric. It is aninteresting case of nonholonomic geometries when torsion is completely trans-formed into nonmetricity which for physicists presented a substantial interestin connection to the Weyl nonmetricity introduced as a method of preservingconformal symmetry of certain scalar field constructions in general relativity,see discussion in [22]. Nevertheless, it should be noted that the constructionswith the Chern connection, in general, are not metric compatible and cannot beapplied in direct form to standard models of physics.

It should be noted that all mentioned types of d-connections are uniquely definedby the coefficients of Finsler type d-metric and N-connection structure (equivalently,by the coefficients of corresponding generic off-diagonal metric of type (18)) follow-ing well defined geometric conditions. From such d-connections, we can always“extract” the Levi–Civita connection, using formulas of type (37) and (38), andwork in “non-adapted” (to N-connection) form. From a geometric point of view, wecan work with all types of Finsler connections and elaborate equivalent approacheseven different connections have different merits in some directions of physics. Forinstance, in [37,38], there are considered the Kawaguchi metrization procedure andthe Miron’s method of computing all metric compatible Finsler connections startingwith a canonical one. It was analyzed also the problem of transforming one Finslerconnection into different ones on tangent bundles and the formalism of mutualtransforms of connections was reconsidered for nonholonomic manifolds, see detailsin [77].

Different models of Finsler spaces can be elaborated in explicit form for differenttypes of d-metrics, N-connections and d-connections. For instance, for a Finsler Hes-sian (48) defining a particular case of d-metrics (47), or (16), denoted Fg, for anytype of connection (for instance, canonical d-connection, Cartan–Finsler, Berwald,


494 S. I. Vacaru

Chern, etc.), we can compute the curvatures by using formulas (44) when “hat”labels are changed into the corresponding ones “C,B,Ch, . . ..” This way, we modelFinsler geometries on tangent bundles, like it is considered in the bulk of mono-graphs [8, 10, 15, 33, 37, 38, 53], or on nonholonomic manifolds [11, 23, 77, 80–82].

With the aim to develop new applications in standard models of physics, let’s sayin classical general relativity, when Finsler-like structures are modeled on a (pseudo)Riemannian manifold (we shall consider explicit examples in the next sections), it ispositively sure that the canonical Finsler and Cartan connections, and their variantsof canonical d-connection on vector bundles and nonholonomic manifolds, shouldbe preferred for constructing new classes of Einstein spaces and defining certain lowenergy limits to locally anisotropic string gravity models. Here we note that it isa very difficult problem to define Finsler–Clifford spaces with Finsler spinors, non-commutative generalizations to supersymmetric/noncommutative Finsler geometryif we work with nonmetric d-connections, see discussions in [62, 76, 77].

We cite a proof [8] that any Lagrange fundamental function L can be modeled asa singular case for a certain class of Finsler geometries of extra dimension (perhaps,the authors were oriented to prove from a mathematical point of view that it is notnecessary to develop Finsler geometry as a new theory for Lagrange spaces, or theirdual constructions for the Hamilton spaces). This idea, together with the method ofKawaguchi–Miron transforms of connections, can be related to the Poincare philo-sophical concepts about conventionality of the geometric space and field interactiontheories [49,50]. According to the Poincare’s geometry–physics dualism, the proce-dure of choosing a geometric arena for a physical theory is a question of conveniencefor researches to classify scientific data in an economical way, but not an action tobe verified in physical experiments: as a matter of principle, any physical theorycan be equivalently described on various types of geometric spaces by using moreor less “simple” geometric objects and transforms.

Nevertheless, the modern physics paradigm is based on the ideas of objectivereality of physical laws and their experimental and theoretical verifications, at leastin indirect form. The concept of Lagrangian is a very important geometrical andphysical one and we shall distinguish the cases when we model a Lagrange or aFinsler geometry. A physical or mechanical model with a Lagrangian is not only a“singular” case for a Finsler geometry but reflects a proper set of concepts, funda-mental physical laws and symmetries describing real physical effects. We use theterms Finsler and Lagrange spaces in order to emphasize that they are differentboth from geometric and physical points of view. Certain geometric concepts andmethods (like the N-connection geometry and nonholonomic frame transforms . . . )are very important for both types of geometries, modeled on tangent bundles or onnonholonomic manifolds. This will be noted when we use the term Finsler–Lagrangegeometry (structures, configurations, spaces).

One should emphasize that the author of this review should not be consideredas a physicist who does not accept nonmetric geometric constructions in modern



physics. For instance, Part I in monograph [77] is devoted to a deep study ofthe problem when generalized Finsler–Lagrange structures can be modeled onmetric-affine spaces, even as exact solutions in gravity with nonmetricity [22], and,inversely, the Lagrange-affine and Finsler–affine spaces are classified by nonholo-nomic structures on metric-affine spaces. It is a question of convention on the typeof physical theories one models by geometric methods. The standard theories ofphysics are formulated for metric compatible geometries, but further developmentsin quantum gravity may request certain type of nonmetric Finsler-like geometries,or more general constructions. This is a topic for further investigations.

3.1.3. Generalized Lagrange spaces

There are various applications in optics of nonhomogeneous media and gravity (see,for instance, [19,38,64,77]) considering metrics of type gij ∼ eλ(x,y)Lgij(x, y) whichcannot be derived directly from a mechanical Lagrangian. The ideas and methods towork with arbitrary symmetric and non-degenerated tensor fields gij(x, y) were con-cluded in geometric and physical models for generalized Lagrange spaces, denotedGLn = (M, gij(x, y)), on TM , see [37, 38], where gij(x, y) is called the funda-mental tensor field. Of course, the geometric constructions will be equivalent ifwe shall work on N-anholonomic manifolds Vn+n with nonholonomic coordinatesy. If we prescribe an arbitrary N-connection Na

i (x, y) and consider that a metricgij defines both the h- and -v-components of a d-metric (16), we can introducethe canonical d-connection (43) and compute the components of d-curvature (44),define Ricci and Einstein tensors, elaborate generalized Lagrange models of gravity.

If we work with a general fundamental tensor field gij which cannot be trans-formed into Lgij , we can consider an effective Lagrange function,j L(x, y) gab(x, y)yayb and use

Lgab 12

∂2L∂ya∂yb

(50)

as a Lagrange Hessian (47). A space GLn = (M, gij(x, y)) is said to be with aweakly regular metric if Ln = [M,L =

√|L|)] is a Lagrange space. For such spaces,

we can define a canonical nonlinear connection structure

LNaj (x, y) ∂LGa

∂yj,

LGa =14Lgab

(yk

∂L∂yb∂xk

− ∂L∂xa

)=

14Lgab

(∂gbc∂yd

+∂gbd∂yc

− ∂gcd∂yb

)ycyd,

(51)

which allows us to write LNaj in terms of the fundamental tensor field gij(x, y). The

geometry of such generalized Lagrange spaces is completely similar to that of the

jIn [37,38], it is called the absolute energy of a GLn-space, but for further applications in moderngravity the term “energy” may result in certain type of ambiguities.


496 S. I. Vacaru

usual Lagrange one, with that difference that we do not start with a Lagrangianbut with a fundamental tensor field.

In our papers [1, 67], we considered nonholonomic transforms of a metricLga′b′(x, y)

gab(x, y) = e a′

a (x, y)e b′

b (x, y)Lga′b′(x, y) (52)

when Lga′b′ 12 (ea′eb′L+ eb′ea′L) = 0ga′b′ , for ea′ = eaa′(x, y)

∂∂ya , where 0ga′b′ are

constant coefficients (or in a more general case, they should result in a constantmatrix for the d-curvatures (31) of a canonical d-connection (36)). Such construc-tions allowed to derive proper solitonic hierarchies and bi-Hamilton structures forany (pseudo) Riemannian or generalized Finsler–Lagrange metric. The point was towork not with the Levi–Civita connection (for which the solitonic equations becamevery cumbersome) but with a correspondingly defined canonical d-connection allow-ing to apply well defined methods from the geometry of nonlinear connections.Having encoded the “gravity and geometric mechanics” information into solitonichierarchies and convenient d-connections, the constructions were shown to hold trueif they are “inverted” to those with usual Levi–Civita connections.

3.2. An ansatz for constructing exact solutions

We consider a four dimensional (4D) manifold V of necessary smooth class andconventional splitting of dimensions dimV = n+m for n = 2 and m = 2. The localcoordinates are labeled in the form uα = (xi, ya) = (x1, x2, y3 = v, y4), for i = 1, 2and a, b, . . . = 3, 4.

The ansatz of type (16) is parametrized in the form

g = g1(xi)dx1 ⊗ dx1 + g2(xi)dx2 ⊗ dx2

+ h3(xk, v) δv ⊗ δv + h4(xk, v) δy ⊗ δy,

δv = dv + wi(xk, v)dxi, δy = dy + ni(xk, v)dxi (53)

with the coefficients defined by some necessary smooth class functions of type g1,2 =g1,2(x1, x2), h3,4 = h3,4(xi, v), wi = wi(xk, v), ni = ni(xk, v). The off-diagonal termsof this metric, written with respect to the coordinate dual frame duα = (dxi, dya),can be redefined to state a N-connection structure N = [N3

i = wi(xk, v), N4i =

ni(xk, v)] with a N-elongated co-frame (14) parametrized as

e1 = dx1, e2 = dx2, e3 = δv = dv + widxi, e4 = δy = dy + nidx

i. (54)

This vielbein is dual to the local basis

ei =∂

∂xi− wi(xk, v)

∂

∂v− ni(xk, v)

∂

∂y5, e3 =

∂

∂v, e4 =

∂

∂y5, (55)

which is a particular case of the N-adapted frame (13). The metric (53) does notdepend on variable y4, i.e. it possesses a Killing vector e4 = ∂/∂y4, and distinguishthe dependence on the so-called “anisotropic” variable y3 = v.



Computing the components of the Ricci and Einstein tensors for the metric (53)and canonical d-connection (see details on tensors components’ calculus in [68,77]),one proves that the Einstein equations (39) for a diagonal with respect to (54) and(55) source,

Υαβ + ZΥα

β = [Υ11 = Υ2(xi, v),Υ2

2 = Υ2(xi, v),Υ33 = Υ4(xi),Υ4

4 = Υ4(xi)] (56)

transform into this system of partial differential equations:

R11 = R2

2 =1

2g1g2

[g•1g

•2

2g1+

(g•2)2

2g2− g••2 +

g′1g′2

2g2+

(g′1)2

2g1− g′′1

]= −Υ4(xi), (57)

S33 = S4

4 =1

2h3h4[h∗4(ln

√|h3h4|)∗ − h∗∗4 ] = −Υ2(xi, v), (58)

R3i = −wiβ

2h4− αi

2h4= 0, (59)

R4i = − h3

2h4[n∗∗i + γn∗

i ] = 0, (60)

where, for h∗3,4 = 0,

αi = h∗4∂iφ, β = h∗4 φ∗, γ =

3h∗42h4

− h∗3h3, (61)

φ = ln |h∗3/√|h3h4||, (62)

when the necessary partial derivatives are written in the form a• = ∂a/∂x1,a′ = ∂a/∂x2, a∗ = ∂a/∂v. In the vacuum case, we must consider Υ2,4 = 0. Wenote that we use a source of type (56) in order to show that the anholonomicframe method can be applied also for non-vacuum solutions, for instance, whenΥ2 = λ2 = const and Υ4 = λ4 = const, defining locally anisotropic configurationsgenerated by an anisotropic cosmological constant, which in its turn, can be inducedby certain ansatz for the so-calledH-field (absolutely antisymmetric third rank ten-sor field) in string theory [64,68,77]. Here we note that the off-diagonal gravitationalinteractions can model locally anisotropic configurations even if λ2 = λ4, or bothvalues vanish.

In string gravity, the nontrivial torsion components and source κΥαβ can berelated to certain effective interactions with the strength (torsion)

Hµνρ = eµBνρ + eρBµν + eνBρµ

of an antisymmetric field Bνρ, when

Rµν = −14H νρµ Hνλρ (63)

DλHλµν = 0, (64)

see details on string gravity, for instance, in [18, 51]. The conditions (63) and (64)are satisfied by the ansatz

Hµνρ = Zµνρ + Hµνρ = λ[H]

√| gαβ |ενλρ (65)


498 S. I. Vacaru

where ενλρ is completely antisymmetric and the distorsion (from the Levi–Civitaconnection) and Zµαβeµ = eβTα−eαTβ+ 1

2 (eαeβTγ)eγ is defined by the torsiontensor (28). Our H-field ansatz is different from those already used in string gravitywhen Hµνρ = λ[H]

√| gαβ |ενλρ. In our approach, we define Hµνρ and Zµνρ from the

respective ansatz for the H-field and nonholonomically deformed metric, computethe torsion tensor for the canonical distinguished connection and, finally, define the“deformed” H-field as Hµνρ = λ[H]

√| gαβ |ενλρ − Zµνρ.

Summarizing the results for an ansatz (53) with arbitrary signatures εα =(ε1, ε2, ε3, ε4) (where εα = ±1) and h∗3 = 0 and h∗4 = 0, one proves [64, 68, 77]that any off-diagonal metric

g = eψ(xi)[ε1 dx1 ⊗ dx1 + ε2 dx2 ⊗ dx2] + ε3h

20(xi)

× [f∗(xi, v)]2|ς(xi, v)| δv ⊗ δv + ε4[f(xi, v) − f0(xi)]2 δy4 ⊗ δy4,

δv = dv + wk(xi, v)dxk, δy4 = dy4 + nk(xi, v)dxk,

(66)

where ψ(xi) is a solution of the 2D equation ε1ψ•• + ε2ψ′′ = Υ4,

ς(xi, v) = ς[0](xi) −ε38h2

0(xi)

∫Υ2(xk, v)f∗(xi, v)[f(xi, v) − f0(xi)]dv,

for a given source Υ4(xi), and the N-connection coefficients N3i = wi(xk, v) and

N4i = ni(xk, v) are computed following the formulas

wi = −∂iς(xk, v)

ς∗(xk, v)(67)

nk = 1nk(xi) + 2nk(xi)∫

[f∗(xi, v)]2

[f(xi, v) − f0(xi)]3ς(xi, v)dv, (68)

define an exact solution of the system of Einstein equations (57)–(60). It shouldbe emphasized that such solutions depend on arbitrary nontrivial functionsf(xi, v) (with f∗ = 0), f0(xi), h2

0(xi), ς[0](xi), 1nk(xi) and 2nk(xi), and sourcesΥ2(xk, v),Υ4(xi). Such values for the corresponding signatures εα = ±1 have to bedefined by certain boundary conditions and physical considerations. These classesof solutions depending on integration functions are more general than those fordiagonal ansatz depending, for instance, on one radial variable like in the case ofthe Schwarzschild solution (when the Einstein equations are reduced to an effectivenonlinear ordinary differential equation, ODE). In the case of ODE, the integralvarieties depend on integration constants which can be defined from certain bound-ary/asymptotic and symmetry conditions, for instance, from the constraint thatfar away from the horizon the Schwarzschild metric contains corrections from theNewton potential. Because the ansatz (53) results in a system of nonlinear par-tial differential equations (57)–(60), the solutions depend not only on integrationconstants, but on very general classes of integration functions.

The ansatz of type (53) with h∗3 = 0 but h∗4 = 0 (or, inversely, h∗3 = 0 but h∗4 = 0)consist more special cases and request a bit different method of constructing exactsolutions. Nevertheless, such type solutions are also generic off-diagonal and they



may be of substantial interest (the length of paper does not allow to include ananalysis of such particular cases).

A very general class of exact solutions of the Einstein equations with nontrivialsources (56), in general relativity, is defined by the ansatz

g = eψ(xi)[ε1 dx1 ⊗ dx1 + ε2 dx

2 ⊗ dx2]

+ h3(xi, v) δv ⊗ δv + h4(xi, v) δy4 ⊗ δy4,

δv = dv + w1(xi, v)dx1 + w2(xi, v)dx2,

δy4 = dy4 + n1(xi)dx1 + n2(xi)dx2,

(69)

with the coefficients restricted to satisfy the conditions

ε1ψ•• + ε2ψ

′′ = Υ4, h∗4φ/h3h4 = Υ2,

w′1 − w•

2 + w2w∗1 − w1w

∗2 = 0, n′

1 − n•2 = 0,

(70)

for wi = ∂iφ/φ∗, see (62), for given sources Υ4(xk) and Υ2(xk, v). We note that

the second equation in (70) relates two functions h3 and h4 and the third and forthequations satisfy the conditions (42).

Even the ansatz (53) depends on three coordinates (xk, v), it allows us to con-struct more general classes of solutions for d-metrics, depending on four coordinates:such solutions can be related by chains of nonholonomic transforms. New classesof generic off-diagonal solutions will describe nonholonomic Einstein spaces relatedto string gravity, if one of the chain metric is of type (66), or in Einstein gravity, ifone of the chain metric is of type (69).

4. Einstein Gravity and Lagrange–Kahler Spaces

We show how nonholonomic Riemannian spaces can be transformed into almostHermitian manifolds enabled with nonintegrable almost complex structures.

4.1. Almost Hermitian connections and general relativity

We prove that the Einstein gravity on a (pseudo) Riemannian manifold V n+n

can be equivalently redefined as an almost Hermitian model if a nonintegrableN-connection splitting is prescribed. The Einstein theory can also be modified byconsidering certain canonical lifts on tangent bundles. The first class of Finsler–Lagrange like models [71] preserves the local Lorentz symmetry and can beapplied for constructing exact solutions in Einstein gravity or for developing someapproaches to quantum gravity following methods of geometric/deformation quan-tization. The second class of such models [70] can be considered for some extensionsto canonical quantum theories of gravity which can be elaborated in a renormal-izable form, but, in general, result in violation of local Lorentz symmetry by suchquantum effects.


500 S. I. Vacaru

4.1.1. Nonholonomic deformations in Einstein gravity

Let us consider a metric gαβ

(18), which for a (n+n)-splitting by a set of prescribedcoefficients Na

i (x, y) can be represented as a d-metric g (16). Respectively, we canwrite the Einstein equations in the form (7), or, equivalently, in the form (39) withthe source ZΥαβ defined by the off-diagonal metric coefficients of g

αβ, depending

linearly on Nai , and generating the distorsion tensor Z

γαβ .

Computing the Ricci and Einstein d-tensors, we conclude that the Einsteinequations written in terms of the almost Hermitian d-connection can be alsoparametrized in the form (39). Such geometric structures are nonholonomic: work-ing respectively with g, g, we elaborate equivalent geometric and physical modelson V n+n,Vn+n. Even for vacuum configurations, when Υαβ = 0, in the almost Her-mitian model of the Einstein gravity, we have an effective source ZΥαβ induced bythe coefficients of generic off-diagonal metric. Nevertheless, there are possible inte-grable configurations, when the conditions (42) are satisfied. In this case, ZΥαβ = 0,and we can construct effective Hermitian configurations defining vacuum Einsteinfoliations.

One should be noted that the geometry of nonholonomic 2 + 2 splitting in gen-eral relativity, with nonholonomic frames and d-connections, or almost Hermitianconnections, is very different from the geometry of the well known 3 + 1 splittingADM formalism, see [40], when only the Levi–Civita connection is used. Followingthe anholonomic frame method, we work with different classes of connections andframes when some new symmetries and invariants are distinguished and the fieldequations became exactly integrable for some general metric ansatz. Constrain-ing or redefining the integral varieties and geometric objects, we can generate, forinstance, exact solutions in Einstein gravity and compute quantum corrections tosuch solutions.

4.1.2. Conformal lifts of Einstein structures to tangent bundles

Let us consider a pseudo-Riemannian manifold M enabled with a metric gij(x) as asolution of the Einstein equations. We define a procedure lifting gij(x) conformallyon TM and inducing a generalized Lagrange structure and a corresponding almostHermitian geometry. Let us introduce L(x, y) 2(x, y)gab(x)yayb and use

gab 12∂2L∂ya∂yb

(71)

as a Lagrange Hessian for (47). A space GLn = (M,gij(x, y)) possess a weaklyregular conformally deformed metric if Ln = [M,L =

√|L|)] is a Lagrange space.

We can construct a canonical N-connection Nai following formulas (51), using L

instead of L and gab instead of Lgab (50), and define a d-metric on TM ,

g = gij(x, y) dxi ⊗ dxj + gij(x, y)ei ⊗ ej , (72)



where ei = dyi+ N ji dx

i. The canonical d-connection and corresponding curva-tures are constructed as in generalized Lagrange geometry but using g.

For the d-metric (72), the model is elaborated for tangent bundles with holo-nomic vertical frame structure. The linear operator F defining the almost complexstructure acts on TM following formulas F(ei) = −∂i and F(∂i) = ei, whenF F = −I, for I being the unity matrix. The operator F reduces to a complexstructure if and only if the h-distribution is integrable.

The metric g (72) induces a 2-form associated to F following formulasθ(X,Y) g(FX,Y) for any d-vectors X and Y. In local form, we haveθ = gij(x, y)dyi ∧ dxj . The canonical d-connection D, with N-adapted coeffi-cients Γαβγ = ( Lijk,

Cabc), and corresponding d-curvature has to be computed

with e bb = δbb and gij used instead of gij .

The model of almost Hermitian gravity H2n(TM,g,F) can be applied inorder to construct different extensions of general relativity to geometric quantummodels on tangent bundle [70]. Such models will result positively in violation of localLorentz symmetry, because the geometric objects depend on fiber variables ya. Thequasi-classical corrections can be obtained in the approximation ∼ 1. We omitin this work consideration of quantum models, but note that Finsler methods andalmost Kahler geometry seem to be very useful for such generalizations of Einsteingravity.

5. Finsler–Lagrange Metrics in Einstein & String Gravity

We consider certain general conditions when Lagrange and Finsler structures canbe modeled as exact solutions in string and Einstein gravity. Then, we analyze twoexplicit examples of exact solutions of the Einstein equations modeling generalizedLagrange–Finsler geometries and nonholonomic deformations of physically valuableequations in Einstein gravity to such locally anisotropic configurations.

5.1. Einstein spaces modeling generalized Finsler structures

In this section, we outline the calculation leading from generalized Lagrange andFinsler structures to exact solutions in gravity. Let us consider

εg = εgi′j′ (xk′, yl

′)(ei

′ ⊗ ej′+ ei

′ ⊗ ej′), (73)

where εgi′j′ can be any metric defined by nonholonomic transforms (52) or a v-metric Lgij (50), Lgij (45), or F gij (48). The co-frame h- and v-bases

ei′= ei

′i(x, y) dx

i, ea′= ea

′a(x, y) δy

a = ea′a(dy

a + Nai dx

i) = ea′+ N

ai′e

i′ ,

define ea′= ea

′ady

a and Nai′ = ea

′aN

ai′e

i′i, when

εgi′j′ = e ii′ eii′ gij , hab = εga′b′ e

a′aeb′b, N

ai = ηai (x, y)

εNai , (74)

where we do not consider summation on indices for “polarization” functions ηai andεNa

i is a canonical connection corresponding to εgi′j′ .


502 S. I. Vacaru

The d-metric (73) is equivalently transformed into the d-metric

εg = gij(x)dxi ⊗ dxj + hab(x, y)δya ⊗ δya,

δya = dya + Nai (x, y)dxi,

(75)

where the coefficients gij(x), hab(x, y) and Nai (x, y) are constrained to be defined

by a class of exact solutions (66), in string gravity, or (69), in Einstein gravity. If itis possible to get the limit ηai → 1, we can say that an exact solution (75) modelsexactly a respective (generalized) Lagrange, or Finsler, configuration. We argue thatwe define a nonholonomic deformation of a Finsler (Lagrange) space given by dataεgi′j′ and εNa

i as a class of exact solutions of the Einstein equations given by datagij , hab and N

ai , for any ηai (x, y) = 1. Such constructions are possible, if certain

nontrivial values of e ii′ , ea′a and ηai can be algebraically defined from relations (74)

for any defined sets of coefficients of the d-metric (73) and (75).Expressing a solution in the form (73), we can define the corresponding almost

Hermitian 1–formθ = gi′j′ (x, y)ej′ ∧ ei′ , and construct an almost Hermitian geom-

etry characterizing this solution for F(ei′) = −ei′ and F(ei′) = ei′ , when ei′ =e ii′ (

∂∂xi − N

ai

∂∂ya ) = ei′ − N

a′i′ ea′ . This is convenient for further applications to

certain models of quantum gravity and geometry. For explicit constructions of thesolutions, it is more convenient to work with parametrizations of type (75).

Finally, in this section, we note that the general properties of integral varietiesof such classes of solutions are discussed in [66, 77].

5.2. Deformation of Einstein exact solutions into

Lagrange–Finsler metrics

Let us consider a metric ansatz gαβ (16) with quadratic metric interval

ds2 = g1(x1, x2)(dx1)2 + g2(x1, x2)(dx2)2

+h3(x1, x2, v)[dv + w1(x1, x2, v)dx1 + w2(x1, x2, v)dx2]2

+h4(x1, x2, v)[dy4 + n1(x1, x2, v)dx1 + n2(x1, x2, v)dx2]2 (76)

defining an exact solution of the Einstein equations (7), for the Levi–Civita con-nection, when the source Υαβ is zero or defined by a cosmological constant.We parametrize the coordinates in the form uα = (x1, x2, y3 = v, y4) and theN-connection coefficients as N

3i = wi and N

4i = ni.

We nonholonomically deform the coefficients of the primary d-metric (76),similarly to (74), when the target quadratic interval

ds2η = gi(dxi)2 + ha(dya +Nai dx

i)2 = ei′i e

j′jεgi′j′dx

idxj

+ ea′a e

b′bεga′b′(dya + ηai

εNai dx

i)(dyb + ηbjεN b

j dxj) (77)



can be equivalently parametrized in the form

ds2η = ηj gj(xi)(dxj)2

+ η3(xi, v)h3(xi, v)[dv + wηi(xk, v)εwi(xk, v)dxi]2

+η4(xi, v)h4(xi, v)[dy4 + nηi(xk, v) εni(xk, v)dxi]2, (78)

similarly to ansatz (53), and defines a solution of type (66) (with N-connectioncoefficients (67) and (68)), for the canonical d-connection, or a solution of type (69)with the coefficients subjected to solve the conditions (70).

The class of target metrics (77) and (78) defining the result of a nonholonomicdeformation of the primary data [gi, ha, N

bi ] to a Finsler–Lagrange configuration

[εgi′j′ , εN bj ] are parametrized by values ei

′i, e

a′a and ηai . These values can be expressed

in terms of some generation and integration functions and the coefficients of theprimary and Finsler-like d-metrics and N-connections in such a manner when aprimary class of exact solutions is transformed into a “more general” class of exactsolutions. In a particular case, we can search for solutions when the target metricstransform into primary metrics under some infinitesimal limits.

In general form, the solutions of Eqs. (40) transformed into the system of partialdifferential equations (57)–(60), for the d-metrics (77), equivalently (78), are givenby corresponding sets of frame coefficients

e1′1 =

√|η1|

√|g1| × εE+, e2

′1 =

√|η2|

√|g1| /εE+,

e1′2 = −

√|η2|

√|g2| × g1′2′/εE−, e2

′2 =

√|η2|

√|g2| × εE−, (79)

e3′3 =

√|η3|

√|h3| × εE+, e4

′3 =

√|η3|

√|h3| /εE+,

e3′4 = −

√|η4|

√|h4| × g1′2′/εE−, e4

′4 =

√|η4|

√|h4| × εE−, (80)

where εE± =√|εg1′1′εg2′2′ [(εg1′1′)2εg2′2′ ± (εg1′2′)3]−1| and h-polarizations ηj are

defined from gj = ηj gj(xi) = εjeψ(xi), with signatures εi = ±1, for ψ(xi) being a

solution of the 2D equation

ε1ψ•• + ε2ψ

′′ = λ, (81)

for a given source Υ4(xi) = λ, and the v–polarizations ηa defined from the dataha = ηaha, for

h3 = ε3h20(x

i)[f∗(xi, v)]2|λς(xi, v)|, h4 = ε4[f(xi, v) − f0(xi)]2,

λς = ς[0](xi) −ε38λh2

0(xi)

∫f∗(xi, v)[f(xi, v) − f0(xi)]dv,

(82)

for Υ2(xk, v) = λ. The polarizations ηai of N-connection coefficients N3i = wi =

wηi(xk, v)εwi(xk, v), N4i = ni = nηi(xk, v) εni(xk, v) are computed from respective


504 S. I. Vacaru

formulas

wηiεwi = −∂i

λς(xk, v)λς∗(xk, v)

, (83)

nηkεnk = 1nk(xi) + 2nk(xi)

∫[f∗(xi, v)]2λς(xi, v)[f(xi, v) − f0(xi)]3

dv. (84)

We generate a class of exact solutions for Einstein spaces with Υ2 = Υ4 = λ ifthe integral varieties defined by gj, ha, wi and ni are subjected to constraints (70).

5.3. Solitonic pp-waves and their effective Lagrange spaces

Let us consider a d-metric of type (76),

δs2[pw] = −dx2 − dy2 − 2κ(x, y, v) dv2 + dp2/8κ(x, y, v), (85)

where the local coordinates are x1 = x, x2 = y, y3 = v, y4 = p, and the nontrivialmetric coefficients are parametrizedg1 = −1, g2 = −1, h3 = −2κ(x, y, v), h4 =1/8 κ(x, y, v). This is vacuum solution of the Einstein equation defining pp-waves[47]: for any κ(x, y, v) solving κxx + κyy = 0, with v = z + t and p = z − t, where(x, y, z) are usual Cartesian coordinates and t is the time like coordinate. Twoexplicit examples of such solutions are given by κ = (x2 − y2) sin v, defining a planemonochromatic wave, or by κ = xy/(x2 + y2)2 exp[v2

0 − v2], for |v| < v0, and κ = 0,for |v| ≥ v0, defining a wave packet traveling with unit velocity in the negative zdirection.

We nonholonomically deform the vacuum solution (85) to a d-metric of type (78)

ds2η = −eψ(x,y)[(dx)2 + (dy)2]

− η3(x, y, v) · 2κ(x, y, v)[dv + wηi(x, y, v, p)εwi(x, y, v, p)dxi]2

+ η4(x, y, v) ·1

8κ(x, y, v)[dy4 + nηi(x, y, v, p) εni(x, y, v, p)dxi]2, (86)

where the polarization functions η1 = η2 = eψ(x,y), η3,4(x, y, v),wηi(x, y, v) andnηi(x, y, v) have to be defined as solutions in the form (81), (82), (83) and (84) fora string gravity ansatz (65), λ = λ2

H/2, and a prescribed (in this section) analogousmechanical system with

Nai = wi(x, y, v) = wηi

Lwi, ni(x, y, v) = nηiεni (87)

for ε = L(x, y, v, p) considered as regular Lagrangian modeled on a N-anholonomicmanifold with holonomic coordinates (x, y) and nonholonomic coordinates (v, p).

A class of 3D solitonic configurations can defined by taking a polarization func-tion η4(x, y, v) = η(x, y, v) as a solution of solitonic equationk

η•• + ε(η′ + 6η η∗ + η∗∗∗)∗ = 0, ε = ±1, (88)

kAs a matter of principle we can consider that η is a solution of any 3D solitonic, or other,nonlinear wave equation.



and η1 = η2 = eψ(x,y) as a solution of (81) written as

ψ•• + ψ′′ =λ2H

2. (89)

Introducing the above stated data for the ansatz (86) into Eq. (58), we get twoequations relating h3 = η3 h3 and h4 = η4h4,

η4 = 8κ(x, y, v)[h

[0]4 (x, y) +

12λ2

H

e2η(x,y,v)]

(90)

|η3| =e−2η(x,y,v)

2κ2(x, y, v)[(√|η4(x, y, v)|)∗]2, (91)

where h[0]4 (x, y) is an integration function.

Having defined the coefficients ha, we can solve Eqs. (59) and (60) by expressingthe coefficients (61) and (62) through η3 and η4 defined by pp- and solitonic wavesas in (91) and (90). The corresponding solutions

w1 = wη1Lw1 = (φ∗)−1∂xφ, w2 = wη1

Lw1 = (φ∗)−1∂yφ, (92)

ni = n[0]i (x, y) + n

[1]i (x, y)

∫|η3(x, y, v)η−3/2

4 (x, y, v)|dv, (93)

are for φ∗ = ∂φ/∂v, see formulas (62), where n[0]i (x, y) and n[1]

i (x, y) are integrationfunctions. The values eψ(x,y), η3 (91), η4 (90), wi (92) and ni (93) for the ansatz(86) completely define a nonlinear superpositions of solitonic and pp-waves as anexact solution of the Einstein equations in string gravity if there are prescribedsome initial values for the nonlinear waves under consideration. In general, suchsolutions depend on some classes of generation and integration functions.

It is possible to give a regular Lagrange analogous interpretation of an explicitexact solution (86) if we prescribe a regular Lagrangian ε = L(x, y, v, p), with Hes-sian Lgi′j′ = 1

2∂2L

∂yi′∂yj′ , for xi′

= (x, y) and ya′

= (v, p). Introducing the valuesLgi′j′ , η1 = η2 = eψ, η3, η4 and h3, h4, all defined above, into (79) and (80), wecompute the vierbein coefficients ei

′i and ea

′a which allows us to redefine equivalently

the quadratic element in the form (77), for which the N-connection coefficients LNai

(46) are nonholonomically deformed to Nai (87). With respect to such nonholonomic

frames of references, an observer “swimming in a string gravitational ocean of inter-acting solitonic and pp-waves” will see his world as an analogous mechanical modeldefined by a regular Lagrangian L.

5.4. Finsler–solitonic pp-waves in Schwarzschild spaces

We consider a primary quadratic element

δs20 = −dξ2 − r2(ξ) dϑ2 − r2(ξ) sin2 ϑ dϕ2 +2(ξ) dt2, (94)

where the nontrivial metric coefficients are

g2 = −r2(ξ), h3 = −r2(ξ) sin2 ϑ, h4 = 2(ξ), (95)


506 S. I. Vacaru

with x1 = ξ, x2 = ϑ, y3 = ϕ, y4 = t, g1 = −1, ξ =∫dr|1 − 2µ

r |1/2 and 2(r) =1− 2µ

r . For µ being a point mass, the element (94) defines the Schwarzschild solutionwritten in spacetime spherical coordinates (r, ϑ, ϕ, t).

Our aim, is to find a nonholonomic deformation of metric (94) to a class of newvacuum solutions modeled by certain types of Finsler geometries.

The target stationary metrics are parametrized in the form similar to (78), seealso (69),

ds2η = −eψ(ξ,ϑ)[(dξ)2 + r2(ξ)(dϑ)2]

− η3(ξ, ϑ, ϕ) · r2(ξ) sin2 ϑ [dϕ+ wηi(ξ, ϑ, ϕ, t)Fwi(ξ, ϑ, ϕ, t)dxi]2

+ η4(ξ, ϑ, ϕ) ·2(ξ) [dt+ nηi(ξ, ϑ, ϕ, t)Fni(ξ, ϑ, ϕ, t)dxi]2. (96)

The polarization functions η1 = η2 = eψ(ξ,ϑ), ηa(ξ, ϑ, ϕ),wηi(ξ, ϑ, ϕ) andnηi(ξ, ϑ, ϕ) have to be defined as solutions of (70) for Υ2 = Υ4 = 0 and a prescribed(in this section) locally anisotropic, on ϕ, geometry with Na

i = wi(ξ, ϑ, ϕ) =wηi

Fwi, ni(ξ, ϑ, ϕ) = nηiFni, for ε = F 2(ξ, ϑ, ϕ, t) considered as a fundamental

Finsler function for a Finsler geometry modeled on a N-anholonomic manifold withholonomic coordinates (r, ϑ) and nonholonomic coordinates (ϕ, t). We note thateven the values wηi,

Fwi, nηi, and Fni can depend on time like variable t, suchdependencies must result in N-connection coefficients of type Na

i (ξ, ϑ, ϕ).Putting together the coefficients solving the Einstein equations (58)–(60) and

(70), the class of vacuum solutions in general relativity related to (96) can beparametrized in the form

ds2η = −eψ(ξ,ϑ)[(dξ)2 + r2(ξ)(dϑ)2]

− h20 [b∗(ξ, ϑ, ϕ)]2[dϕ+ w1(ξ, ϑ)dξ + w2(ξ, ϑ)dϑ]2

+ [b(ξ, ϑ, ϕ) − b0(ξ, ϑ)]2 [dt+ n1(ξ, ϑ)dξ + n2(ξ, ϑ)dϑ]2, (97)

where h0 = const and the coefficients are constrained to solve the equations

ψ•• + ψ′′

= 0, w′1 − w•

2 + w2w∗1 − w1w

∗2 = 0, n′

1 − n•2 = 0, (98)

for instance, for w1 = (b∗)−1(b + b0)•, w2 = (b∗)−1(b + b0)′, n•2 = ∂n2/∂ξ and

n′1 = ∂n1/∂ϑ.

The polarization functions relating (97) to (96), are computed

η1 = η2 = eψ(ξ,ϑ), η3 = [h0b∗/r(ξ) sinϑ]2, η4 = [(b− b0)/]2,

wηi = wi(ξ, ϑ)/Fwi(ξ, ϑ, ϕ, t), nηi = ni(ξ, ϑ)/Fni(ξ, ϑ, ϕ, t).(99)

The next step is to chose a Finsler geometry which will model (97), equiv-alently (96), as a Finsler-like d-metric (77). For a fundamental Finsler functionF = F (ξ, ϑ, ϕ, t), where xi

′= (ξ, ϑ) are h-coordinates and ya

′= (ϕ, t) are

v-coordinates, we compute F ga′b′ = (1/2)∂2F/∂ya′∂yb

′following formulas (48) and

parametrize the Cartan N-connection as CNai = Fwi, Fni. Introducing the values



(95)F gi′j′ and polarization functions (99) into (79) and (80), we compute the vier-bein coefficients ei

′i and ea

′a which allows us to redefine equivalently the quadratic

element in the form (77), in this case, for a Finsler configuration for which theN-connection coefficients CNa

i (46) are nonholonomically deformed to Nai satisfy-

ing the last two conditions in (98). With respect to such nonholonomic frames ofreferences, an observer “swimming in a locally anisotropic gravitational ocean” willsee the nonholonomically deformed Schwarzschild geometry as an analogous Finslermodel defined by a fundamental Finser function F .

6. Outlook and Conclusions

In this review article, we gave a self-contained account of the core developments ongeneralized Finsler–Lagrange geometries and their modeling on (pseudo) Rieman-nian and Riemann–Cartan manifolds provided with preferred nonholonomic framestructure. We have shown how the Einstein gravity and certain string models ofgravity with torsion can be equivalently reformulated in the language of gener-alized Finsler and almost Hermitian/Kahler geometries. It was also argued thatformer criticism and conclusions on experimental constraints and theoretical dif-ficulties of Finsler-like gravity theories were grounded only for certain classes oftheories with metric noncompatible connections on tangent bundles and/or result-ing in violation of local Lorentz symmetry. We emphasized that there were omittedresults when for some well defined classes of nonholonomic transforms of geometricstructures we can model geometric structures with local anisotropy, of Finsler–Lagrange type, and generalizations, on (pseudo) Riemann spaces and Einsteinmanifolds.

Our idea was to consider not only some convenient coordinate and frame trans-forms, which simplify the procedure of constructing exact solutions, but also todefine alternatively new classes of connections which can be employed to generatenew solutions in gravity. We proved that the solutions for the so-called canonicaldistinguished connections can be equivalently re-defined for the Levi–Civita connec-tion and/or constrained to define integral varieties of solutions in general relativity.

The main conclusion of this work is that we can avoid all existing experimentalrestrictions and theoretical difficulties of Finsler physical models if we work withmetric compatible Finsler-like structures on nonholonomic (Riemann, or Riemann–Cartan) manifolds but not on tangent bundles. In such cases, all nonholonomicconstructions modeled as exact solutions of the Einstein and matter field equa-tions (with various string, quantum field . . . corrections) are compatible with thestandard paradigm in modern physics.

In other turn, we emphasize that in quantum gravity, statistical and thermo-dynamical models with local anisotropy, gauge theories with constraints and bro-ken symmetry and in geometric mechanics, nonholonomic configurations on (co)tangent bundles, of Finsler type and generalizations, metric compatible or withnonmetricity, also seem to be very important.


508 S. I. Vacaru

Various directions in generalized Finsler geometry and applications havematured enough so that some tenths of monographs have been written, includingsome recent and updated: we cite here [2,4–8,10,14,33–39,54,62,76,77]. These mono-graphs approach and present the subjects from different perspectives depending, ofcourse, on the author’s own taste, historical period and interests both in geome-try and physics. The monograph [77] summarizes and develops the results orientedto application of Finsler methods to standard theories of gravity (on nonholonomicmanifolds, not only on tangent bundles) and their noncommutative generalizations;it was also provided a critical analysis of the constructions with nonmetricity andviolations of local Lorentz symmetry.

Finally, we suggest the reader to see a more complete review [72] discussingapplications of Finsler and Lagrange geometry both to standard and nonstandardmodels of physics (presenting a variant which was not possible to be publishedbecause of limit of space in this journal).

Acknowledgments

The work is performed during a visit at the Fields Institute. The author thanks M.Anastasiei, A. Bejancu, V. Obadeanu and V. Oproiu for discussions and providingvery important references on the geometry of Finsler–Lagrange spaces, nonholo-nomic manifolds and related almost Kahler geometry.

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Finsler and Lagrange Geometries in Einstein and String Gravity