-projective geometryh-projective geometry Stefan Rosemann (based on joint works with V. Matveev, A. Fedorova, V. Kiosak)Mathematisches Institut Friedrich-Schiller-Universität Jena

h-projective geometry

Stefan Rosemann(based on joint works with V. Matveev, A. Fedorova, V. Kiosak)

Mathematisches InstitutFriedrich-Schiller-Universität Jena

Oppurg, March 24, 2012

S. Rosemann (FSU Jena) h-projective geometry 1 / 19

Aims of my talk


Aims of my talk

I will try to explain the following:h-projective geometry vs. projective geometry


Aims of my talk


What is h-projective geometry ?


Aims of my talk



What are the main problems ?


Aims of my talk




Our contributions to the main problems.


Aims of my talk





h-projective geometry in PDE-language.


Aims of my talk





h-projective geometry in PDE-language.

Relation to integrable systems.


Motivation: projective geometry

Two Riemannian metrics g and g on a manifold M are calledprojectively equivalent if their unparametrized geodesics coincide.




Otherwise put, ∇γ γ = αγ ⇔ ∇γ γ = αγ





Typical question: Given g, is there a metric g which is projectivelyequivalent to g in a non-trivial way?

(where non-trivial usually means ∇ 6= ∇)







Otsuki, Tashiro 1954: There are only trivial examples ofprojectively equivalent Kähler metrics.







Otsuki, Tashiro 1954: There are only trivial examples ofprojectively equivalent Kähler metrics.

⇒ They suggested a new notion of equivalence:

h-projective equivalence!


h-planar curves

Let (M,g, J) be a Kähler manifold and ∇ the Levi-Civitaconnection of g.


h-planar curves


DefinitionA regular curve γ : I → M is called h-planar with respect to g if

∇γ γ = αγ + βJγ

for some functions α, β : I → R.


h-planar curves


DefinitionA regular curve γ : I → M is called h-planar with respect to g if

∇γ γ = αγ + βJγ

for some functions α, β : I → R.

There are infinitely manyh-planar curves γ withγ(0) = x and γ(0) = ζ for eachx ∈ M and ζ ∈ TxM.

Unparametrized geodesicssatisfy ∇γ γ = αγ.

ζ

xγ


Examples

Cn with Kähler metric

∑ni=1 dz i ⊗ dz i :

A curve γ is h-planar ⇔γ is contained in a complex linespan

C{v} for some v 6= 0.


Examples





(CP(n),gFubini-Study, Jstandard):

By definition a projective line L is the projection of a complex planeE ⊆ Cn+1 onto CP(n).


Examples






By definition a projective line L is the projection of a complex planeE ⊆ Cn+1 onto CP(n).L is a 2-dim complex totally geodesic submanifold of(CP(n), gFS , Jstandard ):


Examples







⇒ For each curve γ : I → L ⊆ CP(n), we have

∇γ(t)γ ∈ Tγ(t)L


Examples








∇γ(t)γ ∈ Tγ(t)L = spanR{γ(t), Jstandard(γ(t))}


Examples









⇒ ∇γ γ = αγ + βJstandard(γ)


Examples









⇒ ∇γ γ = αγ + βJstandard(γ)

We obtain:

A curve γ is h-planar with respect to gFS ⇔γ is contained in aprojective line L


h-projective equivalence

DefinitionLet g and g be two Kähler metrics on the complex manifold (M, J). Wecall g and g h-projectively equivalent if each h-planar curve of g ish-planar with respect to g and vice versa.




Main problem: Given g, does there exist g beeing h-projectivelyequivalent to g in a non-trivial way?





DefinitionA bi-holomorphic mapping f : M → M is called h-projectivetransformation if f ∗g is h-projectively equivalent to g.





DefinitionA bi-holomorphic mapping f : M → M is called h-projectivetransformation if f ∗g is h-projectively equivalent to g.Equivalently, we can require that f preserves the set of h-planar curvesof g.






Denote by HPro(g, J) group of h-projective transformations.






Denote by HPro(g, J) group of h-projective transformations.

Main problem reformulated: When is HPro(g, J) 6= Iso(g, J)?


h-projective equivalence on (CP(n), gFS, Jstandard)

A ∈ Gl(n + 1,C) maps complex lines onto complex lines



A ∈ Gl(n + 1,C) maps complex lines onto complex lines⇒ It induces a bi-holomorphic transformation

fA : CP(n) → CP(n)





Since A maps complex planes onto complex planes we have:






⇒ fA maps projective lines onto projective lines.







⇒ Then fA preserves the set of h-planar curves.








⇒ fA ∈ HPro(gFS, Jstandard ), hence, f ∗AgFS and gFS areh-projectively equivalent.








⇒ fA ∈ HPro(gFS, Jstandard ), hence, f ∗AgFS and gFS areh-projectively equivalent.

Since fA is isometry iff A is proportional to some element inU(n + 1), we have

HPro(gFS, Jstandard ) 6= Iso(gFS, Jstandard ).


Two results on the main problems

Main problem 1: Given g, does there exist g beeing h-projectivelyequivalent to g in a non-trivial way?




Theorem (joint with Fedorova, Kiosak, Matveev,arXiv:1009.5530v1, Proc. London Math. Soc. 2011)Let (M, J) be a compact complex manifold of real dimension 2n ≥ 4.




Theorem (joint with Fedorova, Kiosak, Matveev,arXiv:1009.5530v1, Proc. London Math. Soc. 2011)Let (M, J) be a compact complex manifold of real dimension 2n ≥ 4.Suppose g1,g2,g3 are linearly independent (non-trivial) h-projectivelyequivalent Kähler metrics on (M, J).




Theorem (joint with Fedorova, Kiosak, Matveev,arXiv:1009.5530v1, Proc. London Math. Soc. 2011)Let (M, J) be a compact complex manifold of real dimension 2n ≥ 4.Suppose g1,g2,g3 are linearly independent (non-trivial) h-projectivelyequivalent Kähler metrics on (M, J). Then,(M,gi , J) ∼= (CP(n), ci · gFS, Jstandard ) for certain constants ci 6= 0.





Main problem 2: When is HPro(g, J) 6= Iso(g, J)?





Main problem 2: When is HPro(g, J) 6= Iso(g, J)?

Theorem (joint with Matveev, arXiv:1103.5613v1, 2011)Let (M,g, J) be compact Kähler manifold of real dimension 2n ≥ 4. IfHPro0(g, J) 6= Iso0(g, J) then (M,g, J) ∼= (CP(n), c · gFS, Jstandard ) forcertain constant c > 0.


Is there something else other than(CP(n), gFubini−Study , Jstandard)?



If we do not assume global assumptions such as completeness orcompactness, we have counterexamples to our theorems:




There can be constructed examples of linearly independent(non-trivial) h-projectively equivalent metrics g1, g2, g3 which are notgFubini−Study .





There are examples of Kähler manifolds, not equivalent togFubini−Study , admitting 1-parameter groups of h-projectivetransformations that are not Killing vector fields.






Even in the compact setting there are many Kähler metrics gdifferent from gFubini−Study , admitting a (non-trivial) h-projectivelyequivalent metric:







Apostolov, Gauduchon, Calderbank, Tønnesen-Friedman2004-2008: Local and global classification of Kähler manifoldsadmitting h-projectively equivalent metric.







Apostolov, Gauduchon, Calderbank, Tønnesen-Friedman2004-2008: Local and global classification of Kähler manifoldsadmitting h-projectively equivalent metric.

⇒ Construction of a lot of examples of such manifolds (with veryinteresting curvature properties).


History and relation to other theories

History:During 1960-1980, one of the main research directions in japanese(Obata, Yano) and soviet (Sinyukov, Mikes) differential geometryschools.




Special cases of our "h-projective transformation result" provenbefore:

Japan (Obata, Yano) France (Lichnerowicz) UdSSR (Sinyukov)Yano, Hiramatu 1981: Akbar-Zadeh 1988: Mikes 1978:

compact + compact + Ricci-flat locally symmetricconstant scalar curvature(CP(n), gFS , Jstandard ) HPro0(g, J) = Iso0(g, J) HPro0(g, J) = Iso0(g, J)

or HPro0(g, J) = Iso0(g, J)








Recent developments are due to its relation tointegrable systems (Kiyohara, Topalov)








Recent developments are due to its relation tointegrable systems (Kiyohara, Topalov)

Kähler manifolds with interesting curvature properties such asextremal, weakly-Bochner flat (Apostolov, Calderbank, Gauduchon,Tønnesen-Friedman).


Two equivalent conditions for h-projective equivalence

Let g, g be Kähler metrics on (M, J).




Tashiro 1956: g, g are h-projectively equivalent iff for a certain1-form Φ

∇X Y −∇X Y = Φ(X )Y +Φ(Y )X − Φ(JX )JY − Φ(JY )JX

for all X ,Y ∈ TM.






for all X ,Y ∈ TM.Now define g-symmetric complex non-degenerate (1,1)-tensor

A(g, g) =(

det gdet g

)1

2(n+1)

g−1g







A(g, g) =(

det gdet g

)1

2(n+1)

g−1g

Mikes, Domashev 1978: g, g are h-projectively equivalent iff for acertain vector field Λ the tensor A = A(g, g) satisfies

(∇X A)Y = g(Y ,X )Λ + g(Y ,Λ)X + g(Y , JX )JΛ + g(Y , JΛ)JX

for all X ,Y ∈ TM.







A(g, g) =(

det gdet g

)1

2(n+1)

g−1g

Mikes, Domashev 1978: g, g are h-projectively equivalent iff for acertain vector field Λ the tensor A = A(g, g) satisfies

(∇X A)Y = g(Y ,X )Λ + g(Y ,Λ)X + g(Y , JX )JΛ + g(Y , JΛ)JX

for all X ,Y ∈ TM. Note that necessarily Λ = 14grad traceA.


Topalov’s quadratic integrals / Killing tensors

Let A = A(g, g) be the (1,1)-tensor constructed fromh-projectively equivalent pair g, g.




Topalov 2003:

The (0,2)-tensors Ft , t ∈ R, given by

Ft(u, v) =√

det (A − tId) g((A − tId)−1u, v)

are a family of Killing tensors for g.




Topalov 2003:

The (0,2)-tensors Ft , t ∈ R, given by

Ft(u, v) =√

det (A − tId) g((A − tId)−1u, v)

are a family of Killing tensors for g.

These Killing tensors Ft commute in the (obvious) sence that

{Ft ,Fs} =

2n∑

i=1

(

∂Ft

∂pi

∂Fs

∂xi−

∂Ft

∂xi

∂Fs

∂pi

)

= 0

for all t , s ∈ R, where Ft = Ft(x)ijpipj , Fs = Fs(x)ijpipj are viewedas functions on T ∗M.


The linear integrals / Killing vector fields





Matveev, Rosemann, Calderbank et al:

The vector fields Kt , t ∈ R, given by

Kt = Jgrad√

det (A − tId)

are a family of commuting Killing vector fields for g.




Matveev, Rosemann, Calderbank et al:

The vector fields Kt , t ∈ R, given by

Kt = Jgrad√

det (A − tId)

are a family of commuting Killing vector fields for g.

All the integrals Ft = F ijt pipj and Ks = K i

spi Poisson-commute forall t , s ∈ R:

{Kt ,Ks} = {Kt ,Fs} = {Ft ,Fs} = 0


Size of the family of integrals

Consider the family Ft(u, v) =√

det (A − tId)g((A − tId)−1u, v) ofKilling tensors for g.

Recall: A is g-symmetric and AJ = JA.





Recall: A is g-symmetric and AJ = JA. In every point x ∈ M,A|TxM has (not necessarily different) real eigenvaluesµ1(x) ≤ ... ≤ µn(x) where 2n = dimRM.






dim span{Ft : t ∈ R} = maximal number of different eigenvalues ofA.






dim span{Ft : t ∈ R} = maximal number of different eigenvalues ofA.

For the Killing vector fields we have dim span{Kt : t ∈ R} = numberof non-constant eigenvalues of A.


The non-degenerate case of h-projective equivalence

A pair g, g of h-projectively equivalent metrics is callednon-degenerate if the corresponding solution A = A(g, g) of themain equation has n different non-constant eigenvalues.




Then the families

Kt = Jgrad√

det (A − tId)

andFt =

√

det (A − tId)g((A − tId)−1., .)

of linear and quadratic integrals give us 2n linear independentintegrals in involution




Then the families

Kt = Jgrad√

det (A − tId)

andFt =

√

det (A − tId)g((A − tId)−1., .)

of linear and quadratic integrals give us 2n linear independentintegrals in involution

⇒ The geodesic flow of g is Liouville integrable.


Local classification for dimRM = 4 and twonon-constant eigenvalues ξ1 and ξ2 of solution A

Following Calderbank et al (2004):There are functions F1,F2 of one variable and coordinatesξ1, ξ2, y1, y2, such that locally the Kähler metric g and the solutiona = g(A., .) take the form

g = ξ1−ξ2F1(ξ1)

dξ21 + ξ2−ξ1

F2(ξ2)dξ2

2 + F1(ξ1)ξ1−ξ2

(dy1 + ξ2dy2)2 + F2(ξ2)

ξ2−ξ1(dy1 + ξ1dy2)

2,

a = ξ1ξ1−ξ2F1(ξ1)

dξ21 + ξ2

ξ2−ξ1F2(ξ2)

dξ22 + ξ1F1(ξ1)−ξ2F2(ξ2)

ξ1−ξ2dy2

1

+ ξ1ξ2(ξ2F1(ξ1)−ξ1F2(ξ2))ξ1−ξ2

dy22 + 2 ξ1ξ2(F1(ξ1)−F2(ξ2))

ξ1−ξ2dy1dy2

The complex structure J is given by

Jdξ1 = F1(ξ1)ξ1−ξ2

(dy1 + ξ2dy2), Jdξ2 = F2(ξ2)ξ2−ξ1

(dy1 + ξ1dy2)

Jdy1 = − ξ1F1(ξ1)

dξ1 −ξ2

F2(ξ2)dξ2, Jdy2 = 1

F1(ξ1)dξ1 +

1F2(ξ2)

dξ2


Global classification of the non-degenerate case

Following Kiyohara, Topalov (2011):

Let (M, J) be a compact complex manifold admitting anon-degenerate pair of h-projectively equivalent metrics. Then Mis a toric variety and as such, it is isomorphic to CP(n).


Global classification of the non-degenerate case

Following Kiyohara, Topalov (2011):

Let (M, J) be a compact complex manifold admitting anon-degenerate pair of h-projectively equivalent metrics. Then Mis a toric variety and as such, it is isomorphic to CP(n).

Moreover, the non-degenerate pairs of h-projectively equivalentmetrics on CP(n) are completely classified.


V. S. Matveev, S. Rosemann, "Proof of the Yano-Obata Conjecturefor h-projective transformations", arXiv:1103.5613v1, 2011

A. Fedorova, V. Kiosak, V. S. Matveev, S. Rosemann, "The onlyKähler manifold with degree of mobility ≥ 3 is(CP(n),gFubini−Study)", arXiv:1009.5530v1, 2010, accepted toProc. of London Math. Soc.

P. Topalov, Geodesic Compatibility And Integrability Of GeodesicFlows, Journal of Mathematical Physics 44, no. 2, 913–929, 2003

K. Kiyohara, Two classes of Riemannian manifolds whosegeodesic flows are integrable, Mem. Amer. Math. Soc. 130, no.619, viii+143 pp., 1997

K. Kiyohara, P. Topalov, On Liouville integrability of h-projectivelyequivalent Kähler metrics, Proc. Amer. Math. Soc. 139(2011),231–242.

V. Apostolov, D. Calderbank, P. Gauduchon, Hamiltonian 2-formsin Kähler geometry I: General theory, J. Differential Geom. 73, no.3, 359–412, 2006


Thank you for your attention!


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-projective geometryh-projective geometry Stefan Rosemann (based on joint works with V. Matveev, A. Fedorova, V. Kiosak)Mathematisches Institut Friedrich-Schiller-Universität Jena