Vladimir S. Matveev (Jena)

Vladimir S. Matveev (Jena)

Symmetries of H-projective andprojective structures:

Proof of Lichnerowicz-Obata andYano-Obata conjectures

Def. Let Γ = (Γijk) be a symmetric affine connection on Mn. A

geodesic c : I → M, c : t 7→ x(t) on (M, g) is given in terms ofarbitrary parameter t as solution of

d2xa

dt2+ Γa

bc

dxb

dt

dxc

dt= α(t)

dxa

dt.



d2xa

dt2+ Γa

bc

dxb

dt

dxc

dt= α(t)

dxa

dt.

Better known version of this formula assumes that the parameteris affine (we denote it by s) and reads

d2xa

ds2+ Γa

bc

dxb

ds

dxc

ds= 0.



d2xa

dt2+ Γa

bc

dxb

dt

dxc

dt= α(t)

dxa

dt.


d2xa

ds2+ Γa

bc

dxb

ds

dxc

ds= 0.

Def. Two connections Γ and Γ on M are projectively equivalent,if every Γ-geodesic is a Γ geodesic.



d2xa

dt2+ Γa

bc

dxb

dt

dxc

dt= α(t)

dxa

dt.


d2xa

ds2+ Γa

bc

dxb

ds

dxc

ds= 0.

Def. Two connections Γ and Γ on M are projectively equivalent,if every Γ-geodesic is a Γ geodesic.

Fact (Levi-Civita 1896); the proof is a simple linear

algebra: The condition that Γ and Γ are projectively equivalent isequivalent to the existence of a one form φ on M such thatΓa

bc = Γabc + δa

bφc + δacφb

Assume now that M2n carries a complex structure J

Def. Let Γ = (Γijk) be a symmetric affine connection on (M2n, J)

compatible w.r.t J (i.e., ∇J = 0. )



compatible w.r.t J (i.e., ∇J = 0. ) An h-planar curve c : I → M,c : t 7→ x(t) on (M, g) is given as solution of

d2xa

dt2 + Γabc

dxb

dtdxc

dt= α(t) dxa

dt+ β(t) dxk

dtJak

(= (α(t) + i · β(t)) · dxdt

.)

In literature, h-planar curves are also called complex geodesics.



compatible w.r.t J (i.e., ∇J = 0. ) An h-planar curve c : I → M,c : t 7→ x(t) on (M, g) is given as solution of

d2xa

dt2 + Γabc

dxb

dtdxc

dt= α(t) dxa

dt+ β(t) dxk

dtJak

(= (α(t) + i · β(t)) · dxdt

.)

In literature, h-planar curves are also called complex geodesics.

◮ ∃ infinitely many h-planarcurves γ with γ(0) = x andγ(0) = ζ for each x ∈ M andζ ∈ TxM.

ζ

xγ

◮ reparameterized geodesics satisfy ∇γ γ = αγ.

d2xa

dt2 + Γabc

dxb

dtdxc

dt= α(t) dxa

dt+ β(t) dxk

dtJak

(= (α(t) + i · β(t)) · dxdt

.)

Def. Two connections Γ and Γ on M are h−projectively equivalent, ifevery Γ-h-planar curve is a Γ- h−planar curve.

d2xa

dt2 + Γabc

dxb

dtdxc

dt= α(t) dxa

dt+ β(t) dxk

dtJak

(= (α(t) + i · β(t)) · dxdt

.)

Def. Two connections Γ and Γ on M are h−projectively equivalent, ifevery Γ-h-planar curve is a Γ- h−planar curve.

Fact (T. Otsuki, Y. Tashiro 1954; the proof is a linear algebra:) Thecondition that Γ and Γ are h-projectively is equivalent to the existence ofa one form φ on M such thatΓa

bc = Γabc + δa

bφc + δacφb − Ja

bφkJkc − Ja

c φkJkb .

Def. A projective structure is the equivalence class of symmetric affineconnections w.r.t. projective equivalence.


Algebraic reformulation of this geometric condition: two (symmetric)connections Γa

bc and Γabc belong to the same equivalence class, if for a

certain 1-form φa we have

Γabc = Γa

bc + δabφc + δa

cφb.





Γabc = Γa

bc + δabφc + δa

cφb.

Def. An h-projective structure on (M, J) is the equivalence class ofsymmetric affine connections w.r.t. h-projective equivalence.





Γabc = Γa

bc + δabφc + δa

cφb.





Γabc = Γa

bc + δabφc + δa

cφb − JabφkJ

kc − Ja

c φkJkb .





Γabc = Γa

bc + δabφc + δa

cφb.





Γabc = Γa

bc + δabφc + δa

cφb − JabφkJ

kc − Ja

c φkJkb .

(This algebraic reformulation does not requite that Γ is compatible withJ. In the case Γ is compatible with J, the connection Γ is alsoautomatically compatible with J.)

History

History

Projective differential geometry is a very classical topic (at least underthe assumption that the connections Γ and Γ are Levi-Civitaconnnections): the first examples are due to Lagrange 1779 and manyquestions were posed and solved (and sometimes remain unsolved) byclassics of differential geometry and mechanics: Beltrami, Dini,Levi-Civita, Painleve, Weyl,...

History


The people in the h−projective geometry, as a rule, haveprojective-geometry as a background: there is one recent exception fromthis rule that will be discussed below.

Example: The founders (1953) of h-projective geometry T. Otsuki, Y.Tashiro worked in the projective geometry before. They defineh-projectively equivalent metrics, because in the Kahler situationprojective equivalence is not interesting.

History


The people in the h−projective geometry, as a rule, haveprojective-geometry as a background: there is one recent exception fromthis rule that will be discussed below.

Example: The founders (1953) of h-projective geometry T. Otsuki, Y.Tashiro worked in the projective geometry before. They defineh-projectively equivalent metrics, because in the Kahler situationprojective equivalence is not interesting.

Most questions studied in the projective geometry can be generalised tothe h-projective setting – and these is what most people before approx.2003 did with the hope that they can also generalize the proofs.

Around 2003 two strong teams reinventedh-projective geometry in completely different terms:

Kiyohara - Topalov: What they called “typ A Kahler-Liouvillesystems” is a special case of h-projectively equivalent metrics.

Apostolov-Calderbank-Gauduchon: What they called“Hamiltonian 2-forms” is precisely the same as h-projectivelyequivalent metrics.

Around 2003 two strong teams reinventedh-projective geometry in completely different terms:

Kiyohara - Topalov: What they called “typ A Kahler-Liouvillesystems” is a special case of h-projectively equivalent metrics.

Apostolov-Calderbank-Gauduchon: What they called“Hamiltonian 2-forms” is precisely the same as h-projectivelyequivalent metrics.

These groups brought new technigue in the subject: integrablesystems technique, symplectic and Kahler geomety technique, andparabolic geometry technique. These new techniques together caneffectively help to solve the problems stated by classics – I willshow two examples.

Symmetries of the projective and h-projective structures.


Def. A vector field is a symmetry of a projective structure, if it sendsgeodesics to geodesics.


Def. A vector field is a symmetry of a projective structure, if it sendsgeodesics to geodesics.Def. A vector field is a symmetry of a h-projective structure, if it sends

h−planar curves to h-planar curves.



h−planar curves to h-planar curves.Easy Theorem 1. A vector field ∂

∂x1 is a symmetry of a projectivestructure [Γ], iff

Γijk(x

1, ..., xn) = Γijk( x2, ..., xn

︸︷︷︸

no x1−coord.

) + δijφk(x

1, ..., xn) + δikφj(x

1, ..., xn).



h−planar curves to h-planar curves.Easy Theorem 1. A vector field ∂

∂x1 is a symmetry of a projectivestructure [Γ], iff

Γijk(x

1, ..., xn) = Γijk( x2, ..., xn

︸︷︷︸

no x1−coord.

) + δijφk(x

1, ..., xn) + δikφj(x

1, ..., xn).

Easy Theorem 2. A vector field ∂∂x1 is a symmetry of a h-projective

structure [Γ], iff

Γijk(x

1, ..., xn) = Γijk( x2, ..., xn

︸︷︷︸

no x1−coord.

) + δijφk(x

1, ..., xn) + δikφj(x

1, ..., xn)

−J ij φa(x

1, ..., xn)Jak − J i

kφa(x1, ..., xn)Ja

j .

Thus, there is almost no sense to study a symmetry of projectiveor h−projective structures.

I will discuss “metric” h-projective geometry

I assume that the projective (or h-projective) structure containsthe Levi-Civita connection of a (pseudo-)Riemannian metric.


I assume that the projective (or h-projective) structure containsthe Levi-Civita connection of a (pseudo-)Riemannian metric. Thatmeans, I am speaking not about (h-)projectively equivalentconnections, but about (h-)projectively equivalent metrics.


I assume that the projective (or h-projective) structure containsthe Levi-Civita connection of a (pseudo-)Riemannian metric. Thatmeans, I am speaking not about (h-)projectively equivalentconnections, but about (h-)projectively equivalent metrics.

It appeared though that this is convenient to fix a connection andthen look for a metric Levi-Civita connection within the(h-)projective class.

We reformulate this condition as a system of PDE.

Theorem (Eastwood-Matveev 2006; special cases known toLiouville 1889, Sinjukov 1961, Bolsinov-Matveev 2003) TheLevi-Civita connection of g lies in a projective class of a connection Γi

jk if

and only if σab := g ab · det(g)1/(n+1) is a solution of

(∇aσ

bc)− 1

n+1

(∇iσ

ibδca + ∇iσ

icδba

)= 0. (∗)


jk if


(∇aσ

bc)− 1

n+1

(∇iσ

ibδca + ∇iσ

icδba

)= 0. (∗)

Here σab := g ab · det(g)1/(n+1) should be understood as an element ofS2M ⊗ (Λn)

2/(n+1)M. In particular,

∇aσbc =

∂

∂xaσbc + Γb

adσdc + Γcdaσ

bd

︸︷︷︸

Usual covariant derivative

− 2

n + 1Γd

da σbc

︸︷︷︸

addition coming from volume form


jk if


(∇aσ

bc)− 1

n+1

(∇iσ

ibδca + ∇iσ

icδba

)= 0. (∗)

Here σab := g ab · det(g)1/(n+1) should be understood as an element ofS2M ⊗ (Λn)


∇aσbc =

∂

∂xaσbc + Γb

adσdc + Γcdaσ

bd

︸︷︷︸


− 2

n + 1Γd

da σbc

︸︷︷︸


The equations (∗) is a system of(

n2(n+1)2 − n

)

linear PDEs of the first

order on n(n+1)2 unknown components of σ.

Theorem (Matveev-Rosemann 2011/independently Calderbank2011) The Levi-Civita connection of g on (M2n, J) lies in a h-projectiveclass of a connection Γi

jk if and only if σab := g ab · det(g)1/(2n+2) is asolution of

∇aσbc − 1

2n(δb

k∇ℓσℓc + δc

a∇ℓσℓb + Jb

a Jcm∇ℓσ

ℓm + Jca Jb

m∇ℓσℓm) = 0. (∗∗)

Theorem (Matveev-Rosemann 2011/independently Calderbank2011) The Levi-Civita connection of g on (M2n, J) lies in a h-projectiveclass of a connection Γi

jk if and only if σab := g ab · det(g)1/(2n+2) is asolution of

∇aσbc − 1

2n(δb

k∇ℓσℓc + δc

a∇ℓσℓb + Jb

a Jcm∇ℓσ

ℓm + Jca Jb

m∇ℓσℓm) = 0. (∗∗)

Here σab := g ab · det(g)1/(2n+2) should be understood as an element ofS2M ⊗ (Λn)


∇aσbc =

∂

∂xaσbc + Γb

adσdc + Γcdaσ

bd

︸︷︷︸


− 1

n + 1Γd

da σbc

︸︷︷︸


Properties and advantages of the equations (∗) (resp.(∗∗))


1. They are linear PDE systems of finite type (close after twoprolongations). In the projective case, there exists at most(n+1)(n+2)

2 -dimensional space of solutions. In the h-projective case,there exists at most (n + 1)2-dimensional space of solutions.




2. they are projective (resp. h-projective) invariant:they do not depend on the choice of a connectionwithing the projective (resp. h-projective) class.




2. they are projective (resp. h-projective) invariant:they do not depend on the choice of a connectionwithing the projective (resp. h-projective) class.

Since the equations are of finite type, it is expected that a generic

projective (resp. h-projective structure) does not admit a metric in the

projective (resp. h-projective) class: the expectation is true:

Theorem (Matveev 2011) Almost every︸︷︷︸

will be explained

metric of dimension ≥ 4 is

projectively rigid (i.e., every metric projectively equivalent to g isproportional to g) .


will be explained



What we understand under almost every?


will be explained



What we understand under almost every? We consider the standarduniform C 2−topology: the metric g is ε−close to the metric g in thistopology, if the components of g and their first and second derivativesare ε−close to that of g .


will be explained



What we understand under almost every? We consider the standarduniform C 2−topology: the metric g is ε−close to the metric g in thistopology, if the components of g and their first and second derivativesare ε−close to that of g . ‘Almost every’ in the statement of Theoremabove should be understood as

the set of geodesically ri-gid 4D metrics contains anopen everywhere dense (inC 2-topology) subset.

A r b i t r a r y s m a l l n e i g h b o r h o o d o f g i n t h e s p a c e o f a l l m e t r i c s o n U w i t h C ² t o p o l o g y

Arb i t r a r y me t r i c g

O p e n s u b s e t o f p r o j e c t i v e l y r ig id met r i cs


will be explained








The result survives in dim 3, if we replace the uniform C 2− topology bythe uniform C 3-topology (based on Sinjukov 1954). In dim 2, the result isagain true, if we replace the uniform C 2− topology by the uniformC 6-topology (based on nontrivial calculations of Kruglikov 2009 andBryant–Dunajski–Eastwood 2011).


will be explained








The result survives in dim 3, if we replace the uniform C 2− topology bythe uniform C 3-topology (based on Sinjukov 1954). In dim 2, the result isagain true, if we replace the uniform C 2− topology by the uniformC 6-topology (based on nontrivial calculations of Kruglikov 2009 andBryant–Dunajski–Eastwood 2011).A similar result is true for h-projective structures (in this case,

C 2−topology is enough in all dimensions).

Main Theorems

Thus, “most” metrics do not admit projective or h-projective symmetry;but still locally there exist tons of examples of metrics admittingprojective or h-projective symmetry.

Main Theorems


Theorem (“Lichnerowicz-Obata conjecture”, Matveev JDG 2007).Let (M, g) be a compact, connected Riemannian manifold of realdimension n ≥ 2. If (M, g) cannot be covered by (Sn, c · ground) forsome c > 0, then Iso 0 = Pro 0.

Main Theorems


Theorem (“Lichnerowicz-Obata conjecture”, Matveev JDG 2007).Let (M, g) be a compact, connected Riemannian manifold of realdimension n ≥ 2. If (M, g) cannot be covered by (Sn, c · ground) forsome c > 0, then Iso 0 = Pro 0.

Theorem (“Yano-Obata conjecture”, Matveev-Rosemann JDG (toappear in 2013), +Fedorova + Kiosak PLM 2012). Let (M, g , J) bea compact, connected Riemannian Kahler manifold of real dimension2n ≥ 4. If (M, g , J) is not (CP(n), c · gFS, Jstandard) for some c > 0, thenIso 0 = HPro 0.(Here gFS is the Fubini-Studi metric).

Special cases were proved before by French, Japanese andSoviet geometry schools.

Lichnerowciz-Obata conjectureFrance(Lichnerowicz)

Japan(Yano, Obata, Tanno)

Soviet Union(Raschewskii)

Couty (1961) provedthe conjecture assu-ming that g is Einsteinor Kahler

Yamauchi (1974) pro-ved the conjecture as-suming that the scalarcurvature is constant

Solodovnikov (1956)proved the conjectureassuming that all ob-jects are real analyticand that n ≥ 3.

Special cases were proved before by French, Japanese andSoviet geometry schools.

Lichnerowciz-Obata conjectureFrance(Lichnerowicz)

Japan(Yano, Obata, Tanno)

Soviet Union(Raschewskii)

Couty (1961) provedthe conjecture assu-ming that g is Einsteinor Kahler

Yamauchi (1974) pro-ved the conjecture as-suming that the scalarcurvature is constant

Solodovnikov (1956)proved the conjectureassuming that all ob-jects are real analyticand that n ≥ 3.

Yano-Obata conjecture

Japan (Obata, Yano) France (Lichnerowicz) USSR (Sinjukov)

Yano, Hiramatu 1981: Akbar-Zadeh 1988: Mikes 1978:constant scalar curvature Ricci-flat locally symmetric

In Sn and CP(n) the groups of projective resp. h-projectivetransformations are much bigger than the groups ofisometries.


We consider the standard Sn ⊂ Rn+1 with the induced metric.



Fact. Geodesics of the sphere are thegreat circles, that are the intersec-tions of the 2-planes containing thecenter of the sphere with the sphere.




Proof. We consider the reflection with respect to the corresponding2-plane. It is an isometry of the sphere; its sets of fixed points is thegreat circle and is totally geodesics.




Proof. We consider the reflection with respect to the corresponding2-plane. It is an isometry of the sphere; its sets of fixed points is thegreat circle and is totally geodesics. Indeed,

would a geodesic tangent to the greatcircle leave it, it would give a contra-diction with the uniqueness theoremfor solutions of ODEd2xa

dt2 + Γabc

dxb

dtdxc

dt= α(t) dxa

dt

(with any fixed α).

I f t h i s i s geodes ic

G r e a t e c i r c l ethen i s t re f l ec t i on i s a l so a geodes i c c o n t r a d i c t i n g t h e u n i q e n c e

Beltrami example

Beltrami example

Beltrami (1865) observed:

Beltrami example


For every A ∈ SL(n + 1)we construct−−−−−−−−→ a : Sn → Sn, a(x) := A(x)

|A(x)|

Beltrami example



|A(x)|

◮ a is a diffeomorphism

Beltrami example



|A(x)|


◮ a takes great circles (geodesics) to great circles (geodesics)

Beltrami example



|A(x)|



◮ a is an isometry iff A ∈ O(n + 1).

Beltrami example



|A(x)|



◮ a is an isometry iff A ∈ O(n + 1).

Thus, Sl(n + 1) acts by projective transformations on Sn. We see thatProj0 is bigger than Iso0 = SO(n + 1)

In CP(n), the situation is essentially the same

Fact. A curve on (CP(n), gFS , J) is h−planar, if and only if it lieson a projective line (which are totally geodesic complex surfaces inCP(n) homeomorphic to the sphere).


Fact. A curve on (CP(n), gFS , J) is h−planar, if and only if it lieson a projective line (which are totally geodesic complex surfaces inCP(n) homeomorphic to the sphere).Proof. For every projective line, there exists an isometry of CP(n)whose space of fixed points is our projective line. Then, everyh-planar curve whose tangent vector is tangent to a projective linestays on the projective line by the uniqueness of the solutions of asystem of ODE d2xa

dt2 + Γabc

dxb

dtdxc

dt= α(t)dxa

dt+ β(t)dxk

dtJak (for fixed

α, β).



dt2 + Γabc

dxb

dtdxc

dt= α(t)dxa

dt+ β(t)dxk

dtJak (for fixed

α, β).From the other side, since the tangent space TL ⊂ TCP(n) ofevery projective line is J−invariant, every curve lying on theprojective line is h-planar ( because ∇γ γ ∈ TL and is therefore alinear combination of γ and J(γ) since TL is two-dimensional andJ-invariant).



dt2 + Γabc

dxb

dtdxc

dt= α(t)dxa

dt+ β(t)dxk

dtJak (for fixed

α, β).From the other side, since the tangent space TL ⊂ TCP(n) ofevery projective line is J−invariant, every curve lying on theprojective line is h-planar ( because ∇γ γ ∈ TL and is therefore alinear combination of γ and J(γ) since TL is two-dimensional andJ-invariant).

Corollary (h-projective analog of Beltrami Example). Thegroup of h-projective transformations is SL(n + 1, C) and is muchbigger than the group of isometries which is SU(n + 1).

Question. The main results and, actually, most questions askedby classics, do not really require projective and h-projectivestructures (since all the questions are about metrics). Why weintroduced them?

Question. The main results and, actually, most questions askedby classics, do not really require projective and h-projectivestructures (since all the questions are about metrics). Why weintroduced them?

Answer. Because we need them in the proof.

Plan of the proof.

Setup. Our manifold is closed and Riemannian. The projective (resp.h-projective) structure of the metric admits a infinitesimal symmetry, i.e.,a vector field v whose flow preserves the projective (resp. h-projective)structure. Our goal is to show that this vector field is a Killing vectorfield unless g has constant sectional curvature (resp. constantholomorphic sectional curvature).

Plan of the proof.


Def. The degree of the mobility of the projective (resp. h-projective)structure [Γ] is the dimension of the space of solutions of the equation(∗) (resp. (∗∗)).

Plan of the proof.


Def. The degree of the mobility of the projective (resp. h-projective)structure [Γ] is the dimension of the space of solutions of the equation(∗) (resp. (∗∗)).The proof depends on the degree of mobility of the projective (resp.h-projective) structure.

If the degree of mobility of the projective structure is 1, everytwo projective (h-projectively, resp.) metrics are proportional.Then, a projective (h−projective) vector field is a infinitesimalhomothety. Since our manifold is closed, every homothety isisometry so our vector field is a Killing.


If the degree of mobility is at least three, then the following(nontrivial) theorem works.

Theorem (Follows from Matveev 2003/Kiosak-Matveev2010/Matveev-Mounoud 2011 for projective structures;Fedorova-Kiosak-Matveev-Rosemann for h-projectivestructures).If the degree of mobility ≥ 3, the LO and YO conjectures hold(even in the pseudo-Riemannian case).




Remark. The methods of proof are very different from themethods of the next part of my talk. I will touch them if I havetime




Remark. The methods of proof are very different from themethods of the next part of my talk. I will touch them if I havetime

Thus, the only remaining case in when the degree of mobilityis 2

The case degree of mobility =2

Let Sol be the space of solutions of the equation (∗) or (∗∗); it is atwo-dimensional vector space. Let v is a projective (resp. h-projective)vector field.


Let Sol be the space of solutions of the equation (∗) or (∗∗); it is atwo-dimensional vector space. Let v is a projective (resp. h-projective)vector field.Important observation. Lv : Sol → Sol , where Lv is the Lie derivative.


Let Sol be the space of solutions of the equation (∗) or (∗∗); it is atwo-dimensional vector space. Let v is a projective (resp. h-projective)vector field.Important observation. Lv : Sol → Sol , where Lv is the Lie derivative.Proof. The equations (∗) (resp. (∗∗)) are projective (resp. h-projective)

invariant.



invariant. Then, in a coordinate system such that v = ∂∂x1 the

coefficients in the equations do not depend on the x1-coordinate.




coefficients in the equations do not depend on the x1-coordinate. Then,for every solution σij its x1-derivative ∂

∂x1 σij , which is precisely the Lie

derivative, is also a solution, .




coefficients in the equations do not depend on the x1-coordinate. Then,for every solution σij its x1-derivative ∂

∂x1 σij , which is precisely the Lie

derivative, is also a solution, .Thus, in a certain basis σ, σ the Lie derivative is given by the following

matrices (where λ, µ ∈ R):

[Lvσ = λσLv σ = µσ

] [Lvσ = λσ +µσLv σ = −µσ +λσ

] [Lvσ = λσ +σLv σ = λσ

]

We obtained that the derivatives of σ, σ along the flow of v are given by




]

.

We obtained that the derivatives of σ, σ along the flow of v are given by




]

.

Thus, the evolution of the solutions along the flow φt of v are given bythe matrices [

φ∗t σ = eλtσ

φ∗t σ = eµt σ

]

[φ∗

t σ = eλt cos(µt)σ +eλt sin(µt)σφ∗

t σ = −eλt sin(µt)σ +eλt cos(µt)σ

]

[φ∗

t σ = eλtσ +teλt σφ∗

t σ = eλt σ

]

.

We will consider all these three cases separately.

The simplest case is when the evolution is given by

[φ∗



]

.


[φ∗



]

.

Suppose our metrics correspond to the element aσ + bσ.


[φ∗



]

.

Suppose our metrics correspond to the element aσ + bσ.Its evolution is given by

φ∗t (aσ + bσ) = a(eλt cos(µt)σ + eλt sin(µt)σ)

+b(−eλt sin(µt)σ + eλt cos(µt)σ)

= eλt√

a2 + b2(cos(µt + α)σ + sin(µt + α)σ),

where α = arccos(a/(√

a2 + b2)).


[φ∗



]

.




= eλt√



a2 + b2)).Now, we use that the metric is Riemannian. Then, for any point x there

exists a basis in TxM such that σ and σ are given by diagonal matrices:σ = diag(s1, s2, ...) and σ = diag(s1, s2, ...).


[φ∗



]

.




= eλt√




exists a basis in TxM such that σ and σ are given by diagonal matrices:σ = diag(s1, s2, ...) and σ = diag(s1, s2, ...).Then, φ∗

t (aσ + bσ) at this point is also diagonal with the ith elementeλt

√a2 + b2(cos(µt + α)si + sin(µt + α)si ).


[φ∗



]

.




= eλt√




exists a basis in TxM such that σ and σ are given by diagonal matrices:σ = diag(s1, s2, ...) and σ = diag(s1, s2, ...).Then, φ∗

t (aσ + bσ) at this point is also diagonal with the ith elementeλt

√a2 + b2(cos(µt + α)si + sin(µt + α)si ).

Clearly, for a certain t we have that φ∗t (aσ + bσ) is degenerate which

contradicts the assumption,

The proof is is similar when the evolution is given by

[φ∗


t σ = eλt σ

]

.


[φ∗


t σ = eλt σ

]

.

We again suppose that our metrics correspond to the element aσ + bσ.


[φ∗


t σ = eλt σ

]

.

We again suppose that our metrics correspond to the element aσ + bσ.Its evolution is given by

φ∗t (aσ + bσ) = a(eλtσ + eλttσ) + b(eλt σ)

= eλt(aσ + (b + at)σ).


[φ∗


t σ = eλt σ

]

.




We again see that unless a 6= 0 there exists t such that φ∗t (aσ + bσ) is

degenerate which contradicts the assumption.


[φ∗


t σ = eλt σ

]

.




We again see that unless a 6= 0 there exists t such that φ∗t (aσ + bσ) is

degenerate which contradicts the assumption. Now, if a = 0, then g

corresponds to σ and v is its Killing vector field,

The most complicated case is when the evolution is given by the matrix

[φ∗

t σ = eλtσφ∗

t σ = eµt σ

]

. (1)

The complete proof needs technical nontrivial details, I will explain theeffect which proves Yano-Obata conjecture under a certain additionalassumption that will be introduced at the proper place.


[φ∗

t σ = eλtσφ∗

t σ = eµt σ

]

. (1)

The complete proof needs technical nontrivial details, I will explain theeffect which proves Yano-Obata conjecture under a certain additionalassumption that will be introduced at the proper place.We take two elements of Sol corresponding to two Riemannian metrics;they are linear combinations of the basis solutions σ and σ and have theform aσ + bσ, cσ + d σ.


[φ∗

t σ = eλtσφ∗

t σ = eµt σ

]

. (1)

The complete proof needs technical nontrivial details, I will explain theeffect which proves Yano-Obata conjecture under a certain additionalassumption that will be introduced at the proper place.We take two elements of Sol corresponding to two Riemannian metrics;they are linear combinations of the basis solutions σ and σ and have theform aσ + bσ, cσ + d σ.We consider A := (aσ + bσ)(cσ + d σ)−1, this is an one-one tensor

whose all eigenvalues are positive. We take an arbitrary point of M andconsider a basis where the metrics are diagonal; in this basis σ and σ arediagonal as well so that A is also diagonal.


[φ∗

t σ = eλtσφ∗

t σ = eµt σ

]

. (1)


whose all eigenvalues are positive. We take an arbitrary point of M andconsider a basis where the metrics are diagonal; in this basis σ and σ arediagonal as well so that A is also diagonal. Then,

φ∗t A := (aeλtσ + beµt σ)(ceλtσ + deµt σ)−1.


[φ∗

t σ = eλtσφ∗

t σ = eµt σ

]

. (1)




We take an arbitrary point x of M and consider a basis where themetrics are diagonal; in this basis σ and σ are diagonal as well:

σ = diag(s1, ..., s2n), σ = diag(s1, ..., s2n).


[φ∗

t σ = eλtσφ∗

t σ = eµt σ

]

. (1)




We take an arbitrary point x of M and consider a basis where themetrics are diagonal; in this basis σ and σ are diagonal as well:

σ = diag(s1, ..., s2n), σ = diag(s1, ..., s2n).

Then, the eigenvalues a1(t), ..., a2n(t) of φ∗t A at the point x are given by

ai := aeλtsi+beµt si

ceλtsi+deµt si.

Let us consider the functions

ai (t) =aeλtsi + beµt siceλtsi + deµt si

in details.



in details.

Since they are eigenvalues of(1,1)-tensor connecting twometrics,

◮ they must be positiveand

◮ they must be bounded.



in details.




Easy “first semester calcu-lus exercise” shows, then thefunction qualitatively look ason the picture.

b / d

a / c



in details.




Easy “first semester calcu-lus exercise” shows, then thefunction qualitatively look ason the picture.

b / d

a / c

Suppose for simplicity (this is the announced additional assumption!)that A does not have constant eigenvalue equal to b/d . Let us nowexchange b by 0. We obtain a bilinear form such that it is a Riemannianmetrics at the points such that ai 6= b/d . The vector field v is ahomothety of this metric. Then, the metric is flat implying (by theclassical result of Beltrami–Schur and by its natural generalization byOtsuki–Tashiro to the h-projective structures) that the metric g hasconstant curvature.

The case of degree of mobility ≥ 3

Theorem (Solodovnikov 1956 in the projectivecase/Fedorova-Kiosak-Matveev-Rosemann 2012 in the h−projectivecase.)If the degree of mobility of a metric g is ≥ 3, then there exists a constantB such that for every solution σ of (∗) resp. (∗∗) the function f : M → R,

f =(

1det(g)

)1/(n+1) ∑

i,j

gijσij resp. f =

(1

det(g)

)1/(2n+2) ∑

i,j

gijσij

satisfy the following equation

◮ projective case: f,ijk = B (2gij f,k + gik f,j + gjk f,i ) (G).

◮ H-projective case:f,ijk = B

(2gij f,k + gik f,j + gjk f,i + Ja

j f,aJik + Jai f,aJjk

)(T).



f =(

1det(g)

)1/(n+1) ∑

i,j

gijσij resp. f =

(1

det(g)

)1/(2n+2) ∑

i,j

gijσij






)(T).

This is a LOCAL differential geometry.



f =(

1det(g)

)1/(n+1) ∑

i,j

gijσij resp. f =

(1

det(g)

)1/(2n+2) ∑

i,j

gijσij






)(T).

This is a LOCAL differential geometry. It is very nontrivial – we went up

to 5th prolongation to prove this result. One can show though that the

constant B is preserved along geodesics and is therefore a universal

constant.





)(T).

The equations (G) and (T) are famous equations and regularly appearedin the differential geometry: Gallot consider them because of the relationto cone geometry, and Tanno considered it because of the relation tospectral geometry. A lot is known:





)(T).


Theorem (Gallot 1978, Tanno 1978). If a closed Riemannianmanifold admits a nonconstant solution of equation (G) resp. (T), thenthe metric has constant positive curvature resp. constant positiveholomorphic curvature.





)(T).


Theorem (Gallot 1978, Tanno 1978). If a closed Riemannianmanifold admits a nonconstant solution of equation (G) resp. (T), thenthe metric has constant positive curvature resp. constant positiveholomorphic curvature.

The Theorem finishes the proof of the conjectures under the assumptionthat the degree of mobility is ≥ 3.

Philosophy and summary


◮ Most questions and many results from projectively equivalentmetrics could be generalized to the pseudo-Riemannian situation.



◮ Moreover, many “h-projective” proofs are in certain sensegeneralisation of the “projective” proofs




◮ In my talk I have explained one example: Lichnerowicz-Obataand Yano-Obata conjecture.





◮ Many classical results in h-projective geometry are obtain bythe same way






◮ Recently, a new group of methods was applied and appeared towork in the h-projective geometry







◮ Quadratic integrals for the geodesic flows (Topalov 2003)







◮ Quadratic integrals for the geodesic flows (Topalov 2003)◮ Linear integrals for the geodesic flows (in the most general

case: Apostolov et al 2006; special cases were known toDomashev-Mikes 1978 and to Topalov – Kiyohara 2003).









◮ Symplectic, Kahler, parabolic, and other techniques.










◮ How one can use these new methods? What should we try to prove?










◮ How one can use these new methods? What should we try to prove?

◮ TOPIC IS HOT!!!! JOIN!!!!!!

Vladimir S. Matveev (Jena)

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Transcript of Vladimir S. Matveev (Jena)