Vladimir Matveev Jena (Germany) › ~matveev › Datei › presentation_rome.pdfTwo separately developed theories, theory of geodesically equivalent metrics and theory of quadratically

Vladimir MatveevJena (Germany)


Lie, Beltrami and Schouten problems


Lie, Beltrami and Schouten problemswww.minet.uni-jena.de/∼matveev/

Two separately developed theories,


◮ theory of geodesically equivalent metrics and



◮ theory of quadratically integrable Hamiltonian systems andseparations of variables



◮ theory of quadratically integrable Hamiltonian systems andseparations of variablesBenenti-systems, L-systems, cofactor systems,quasi-bi-hamiltonian systems, systems admitting specialconformal Killing tensor



◮ theory of quadratically integrable Hamiltonian systems andseparations of variablesBenenti-systems, L-systems, cofactor systems,quasi-bi-hamiltonian systems, systems admitting specialconformal Killing tensor

study essentially the same object.


◮ theory of geodesically equivalent metrics andLevi-Civita Painleve Eisenhart

◮ theory of quadratically integrable Hamiltonian systems andseparations of variablesBenenti-systems, L-systems, cofactor systems,quasi-bi-hamiltonian systems, systems admitting specialconformal Killing tensorLevi-Civita Painleve Eisenhart



◮ theory of geodesically equivalent metrics andLevi-Civita Painleve Eisenhart Weyl Thomas Douglas HallRashevskii Solodovnikov Yamauchi Aminova Venzi MikesShandra

◮ theory of quadratically integrable Hamiltonian systems andseparations of variablesBenenti-systems, L-systems, cofactor systems,quasi-bi-hamiltonian systems, systems admitting specialconformal Killing tensorLevi-Civita Painleve Eisenhart Benenti Braden IbortMagri Marmo Crampin Sarlet Tondo Saunders CantrijnKolokoltsev Rastelli Chanu Ranada Santander KiyoharaBolsinov Fomenko Kozlov Waalkens Dullin Pucacco





study essentially the same object.We apply methods of one in the other




study essentially the same object.We apply methods of one in the other and obtain the announcedclassical problems

Definitions

Definitions

1. Two metrics (on one manifold) are geodesically equivalent ifthey have the same unparametrized geodesics

Definitions

1. Two metrics (on one manifold) are geodesically equivalent ifthey have the same unparametrized geodesics

2. A vector field is projective w.r.t. a metric, if its flow takes(unparametrized) geodesics to geodesics.

Examples of Lagrange 1789 and of Beltrami 1865


��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

0

f(X)X


��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

0

f(X)X

Radial projection f : S2 → R2


��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

0

f(X)X


takes geodesics of the sphere togeodesics of the plane,


��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

0

f(X)X


takes geodesics of the sphere togeodesics of the plane, becausegeodesics on sphere/plane are in-tersection of plains containing 0with the sphere/plane.


��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

0

f(X)X



Beltrami (1865) modified Lagrange example to constructprojective vector fields of the sphere:


��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

0

f(X)X



Beltrami (1865) modified Lagrange example to constructprojective vector fields of the sphere: He observed:


��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

0

f(X)X




For every A ∈ SL(n + 1)


��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

0

f(X)X




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

|A(x)|


��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

0

f(X)X





|A(x)|

◮ a is a diffeomorphism


��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

0

f(X)X





|A(x)|


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


��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

0

f(X)X





|A(x)|



◮ a is an isometry iff A ∈ O(n + 1).


��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

0

f(X)X





|A(x)|




Thus, Sl(n + 1) acts by geodesic-preserving transformations on Sn,


��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

0

f(X)X





|A(x)|




Thus, Sl(n + 1) acts by geodesic-preserving transformations on Sn, and

its algebra sl(n + 1) can be viewed as the algebra of projective vector

fields

Example of Dini 1869


Theorem (Dini 1869)


Theorem (Dini 1869) The metric(X (x) − Y (y))(dx2 + dy2)

is geodesically equivalent to



is geodesically equivalent to(

1Y (y) −

1X (x)

) (dx2

X (x) + dy2

Y (y)

)

,




1Y (y) −

1X (x)

) (dx2

X (x) + dy2

Y (y)

)

, (if

they have sense).




1Y (y) −

1X (x)

) (dx2

X (x) + dy2

Y (y)

)

, (if

they have sense).

Every two nonproportional geodesically equivalent Riemannianmetrics on the surface




1Y (y) −

1X (x)

) (dx2

X (x) + dy2

Y (y)

)

, (if

they have sense).

Every two nonproportional geodesically equivalent Riemannianmetrics on the surface have this form in a neighbourhood of almostevery point.




1Y (y) −

1X (x)

) (dx2

X (x) + dy2

Y (y)

)

, (if

they have sense).





1Y (y) −

1X (x)

) (dx2

X (x) + dy2

Y (y)

)

, (if

they have sense).


Pseudo-Riemannian case:




1Y (y) −

1X (x)

) (dx2

X (x) + dy2

Y (y)

)

, (if

they have sense).


Pseudo-Riemannian case: Two more series of normal forms:Bolsinov, Pucacco, M∼ 2008

Geometry: Relation with integrable systems


Given g , g on Mn we construct−−−−−−−−→


Given g , g on Mn we construct−−−−−−−−→ L := g−1g


Given g , g on Mn we construct−−−−−−−−→ L := g−1g ·

(det gdet g

) 1n+1



(det gdet g

) 1n+1

∀t ∈ R define−−−−−−−−−→ St := (L − t · Id)−1 · det(L − t · Id)



(det gdet g

) 1n+1

∀t ∈ R define−−−−−−−−−→ St := (L − t · Id)−1 · det(L − t · Id)consider−−−−−→ It :



(det gdet g

) 1n+1

∀t ∈ R define−−−−−−−−−→ St := (L − t · Id)−1 · det(L − t · Id)consider−−−−−→ It : TMn → R,



(det gdet g

) 1n+1

∀t ∈ R define−−−−−−−−−→ St := (L − t · Id)−1 · det(L − t · Id)consider−−−−−→ It : TMn → R, It(ξ) := g(St(ξ), ξ).



(det gdet g

) 1n+1

∀t ∈ R define−−−−−−−−−→ St := (L − t · Id)−1 · det(L − t · Id)consider−−−−−→ It : TMn → R, It(ξ) := g(St(ξ), ξ).Theorem (Topalov, M∼ 1998):If g ∼ g , then, ∀t1, t2 ∈ R, the functions Iti are commuting integrals forthe geodesic flow of g (i.e. for the Hamiltonian H(ξ) := g(ξ, ξ))



(det gdet g

) 1n+1


A function I : TM → R is an integral of the geodesic flow of g ,



(det gdet g

) 1n+1


A function I : TM → R is an integral of the geodesic flow of g , if forevery geodesic γ : R → M



(det gdet g

) 1n+1


A function I : TM → R is an integral of the geodesic flow of g , if forevery geodesic γ : R → M the function I (γ, γ) : R → R is constant in t.



(det gdet g

) 1n+1





(det gdet g

) 1n+1



The family contains n integrals which are functionally independentalmost everywhere, if and only if there exists a point such that theminimal polynomial of g−1g has degree n.



(det gdet g

) 1n+1




There is no problem to introduce potential energy in the picture



(det gdet g

) 1n+1




There is no problem to introduce potential energy in the picture(Bolsinov, M∼ 2003/ Crampin, Scarlet 2003/ Benenti 2004 / Kruglikov,M∼ 2006



(det gdet g

) 1n+1




There is no problem to introduce potential energy in the picture(Bolsinov, M∼ 2003/ Crampin, Scarlet 2003/ Benenti 2004 / Kruglikov,M∼ 2006 ; Bolsinov, Pucacco, M∼ 2008 )



(det gdet g

) 1n+1





There is no problem to quantize the system



(det gdet g

) 1n+1





There is no problem to quantize the system (replace the integrals bycommuting differential operators)



(det gdet g

) 1n+1





There is no problem to quantize the system (replace the integrals bycommuting differential operators) (Topalov, M∼ 2001



(det gdet g

) 1n+1





There is no problem to quantize the system (replace the integrals bycommuting differential operators) (Topalov, M∼ 2001 ; Bolsinov,Pucacco, M∼ 2008)

Plan:

◮ Geometric sense of the integrals

Plan:


◮ One application of geodesic equivalence to integrable systems:

Plan:


◮ One application of geodesic equivalence to integrable systems:Sinjukov-Topalov hierarchy as a way of constructing integrablesystems

Plan:



◮ One application of integrable systems to geodesic equivalence:

Plan:



◮ One application of integrable systems to geodesic equivalence:What closed manifolds admit geodesically equivalent Riemannianmetrics?

Plan:




◮ Combining methods from integrable systems and differentialgeometry: solution of Lie Problem

Plan:




◮ Combining methods from integrable systems and differentialgeometry: solution of Lie Problem (joint with Bryant, Manno) andof Lichnerowicz-Obata conjecture

Plan:




◮ Combining methods from integrable systems and differentialgeometry: solution of Lie Problem (joint with Bryant, Manno) andof Lichnerowicz-Obata conjecture

Symplectic nature of these integrals


(Topalov 1997


(Topalov 1997independently

−−− Tabachnikov 1998 ,



−−− Tabachnikov 1998 , Foulon 1986 ,



−−− Tabachnikov 1998 , Foulon 1986 , Pollicott)



−−− Tabachnikov 1998 , Foulon 1986 , Pollicott)Consider Hamiltonian systems




(N2n, ω,H,XH) and (N2n, ω, H,XH)





and their energy surfaces





and their energy surfacesQ2n−1 := {H(x) = h} and Q2n−1 := {H(x) = h}






Suppose there exists m : Q2n−1 → Q2n−1






Suppose there exists m : Q2n−1 → Q2n−1 such that dm(XH) = λ(x)XH







Then we can construct integrals for XH :







Then we can construct integrals for XH :indeed: consider σ := ω|Q ,







Then we can construct integrals for XH :indeed: consider σ := ω|Q , σ := ω|Q and the pull-back m∗σ.







Then we can construct integrals for XH :indeed: consider σ := ω|Q , σ := ω|Q and the pull-back m∗σ.







Then we can construct integrals for XH :indeed: consider σ := ω|Q , σ := ω|Q and the pull-back m∗σ.Lemma: The flow of XH preserves σ, m∗σ.







Then we can construct integrals for XH :indeed: consider σ := ω|Q , σ := ω|Q and the pull-back m∗σ.Lemma: The flow of XH preserves σ, m∗σ.Proof:







Then we can construct integrals for XH :indeed: consider σ := ω|Q , σ := ω|Q and the pull-back m∗σ.Lemma: The flow of XH preserves σ, m∗σ.Proof: LXH

m∗σ = ıXHd [m∗σ] + d [ıXH

m∗σ]









m∗σ] = 0









m∗σ] = 0 + 0









m∗σ] = 0 + 0 = 0.









m∗σ] = 0 + 0 = 0.Since the forms σ, m∗σ are preserved by the flow,









m∗σ] = 0 + 0 = 0.Since the forms σ, m∗σ are preserved by the flow, a function constructedinvariantly by using these forms must automatically be an integral.









m∗σ] = 0 + 0 = 0.Since the forms σ, m∗σ are preserved by the flow, a function constructedinvariantly by using these forms must automatically be an integral. Sothe coefficients of the characteristic polynomial of one form with respectto the second are integrals.

We can construct many new integrable systems

Given g , g let us construct L as above: L := g−1g


Given g , g let us construct L as above: L := g−1g ·(

det gdet g

) 1n+1

For every (1, 1)-tensor B, define:



det gdet g

) 1n+1

For every (1, 1)-tensor B, define:gB(ξ, η) := g(B(ξ), η)gB(ξ, η) := g(B(ξ), η)



det gdet g

) 1n+1


Theorem (Topalov, Matveev 2001):



det gdet g

) 1n+1


Theorem (Topalov, Matveev 2001): Assume g ∼ g . Then, for everyreal-analytic function F , the metrics gF (L) and gF (L) are geodesicallyequivalent (if they have sense.)



det gdet g

) 1n+1



The example of Beltrami gives us a pair of geodesically equivalentmetrics.



det gdet g

) 1n+1



The example of Beltrami gives us a pair of geodesically equivalentmetrics. If we apply the above Theorem to it for functions F (x) = x andF (x) = x2,



det gdet g

) 1n+1



The example of Beltrami gives us a pair of geodesically equivalentmetrics. If we apply the above Theorem to it for functions F (x) = x andF (x) = x2, we get the metrics of the ellipsoid and of the Poissonspheres.



det gdet g

) 1n+1



The example of Beltrami gives us a pair of geodesically equivalentmetrics. If we apply the above Theorem to it for functions F (x) = x andF (x) = x2, we get the metrics of the ellipsoid and of the Poissonspheres.

Application to topology: motivation:


◮ Beltrami 1865:


◮ Beltrami 1865: La seconda ... generalizzazione ... del nostroproblema, vale a dire: riportare i punti di una superficie sopra un’altra superficie in modo che alle linee geodetiche della primacorrispondano linee geodetiche della seconda.



English Translation: DESCRIBE all geodesically equivalent metrics




◮ locally was done by Dini (1869) for dim 2, Levi-Civita (1896)for dim n (almost everywhere for Riem. case)




◮ locally was done by Dini (1869) for dim 2, Levi-Civita (1896)for dim n (almost everywhere for Riem. case) Bolsinov,Pucacco, M∼ (for dim 2 for pseudo-Riemannian case)





◮ I will answer the topologicall question: what closed manifoldscan admit nonproportional geodesically equivalent Riemannianmetrics











◮ I also know the description of all metrics – too long for this talk

What closed manifolds admit geodesically equivalentRiemannian metrics?


Theorem (Matveev 2008) Suppose M is closed connected. LetRiemannian metrics g and g on M be geodesically equivalent andnonproportional.


Theorem (Matveev 2008) Suppose M is closed connected. LetRiemannian metrics g and g on M be geodesically equivalent andnonproportional. Then, one of the following statements holds:



◮ the manifold is diffeomorphic to a reducible space form:




Mdiffeo≈ Sn/G , where G ⊆ O(n + 1)













is discrete





is discrete





is discrete





is discrete





is discrete

acts freely on Sn





is discrete

acts freely on Sn





is discrete

acts freely on Sn





is discrete

acts freely on Sn

= G1 + G2

,

OR◮ it admits a metric with reducible holonomy group.





is discrete

acts freely on Sn

= G1 + G2

,






is discrete

acts freely on Sn

= G1 + G2

,


Corollary 1 (Topalov, M∼ 2001): A closed orientable surface admittingnonproportional geodesically equivalent metrics is S2 or T 2.





is discrete

acts freely on Sn

= G1 + G2

,



Corollary 2 (M∼ 2003):





is discrete

acts freely on Sn

= G1 + G2

,



Corollary 2 (M∼ 2003): A closed 3-manifold admitting nonproportionalgeodesically equivalent metrics is Lp,q or Seifert manifold with zero Eulernumber.





is discrete

acts freely on Sn

= G1 + G2

,



Corollary 2 (M∼ 2003): A closed 3-manifold admitting nonproportionalgeodesically equivalent metrics is Lp,q or Seifert manifold with zero Eulernumber. (Lp,q are covered by S3, Seifert manifold with zero Euler numberare 3-manifolds admitting metrics with reducible holonomy groups.)





is discrete

acts freely on Sn

= G1 + G2

,



Corollary 2 (M∼ 2003): A closed 3-manifold admitting nonproportionalgeodesically equivalent metrics is Lp,q or Seifert manifold with zero Eulernumber. (Lp,q are covered by S3, Seifert manifold with zero Euler numberare 3-manifolds admitting metrics with reducible holonomy groups.)

Proof of Corollary 1:


(Two geodesically equivalent metrics on the surface if genus ≥ 2are proportional)



In dimension 2, the integral I0 is




I0(ξ) :=

(det(g)

det(g)

) 23

g(ξ, ξ).




I0(ξ) :=

(det(g)

det(g)

) 23

g(ξ, ξ).

Assume the surface is neither torus nor the sphere.




I0(ξ) :=

(det(g)

det(g)

) 23

g(ξ, ξ).

Assume the surface is neither torus nor the sphere. The goal is toshow that g and g are proportional.




I0(ξ) :=

(det(g)

det(g)

) 23

g(ξ, ξ).


Because of topology, there exists x0 such that g|x0= g|x0

.




I0(ξ) :=

(det(g)

det(g)

) 23

g(ξ, ξ).


Because of topology, there exists x0 such that g|x0= g|x0

. Weassume g|x1

6= g|x1and find a contradiction.

Explanation of Corollary 2:

Explanation of Corollary 2: (A closed 3-manifold admittingnonproportional geodesically equivalent metrics is Lp,q or Seifertmanifold with zero Euler number.)

Explanation of Corollary 2: (A closed 3-manifold admittingnonproportional geodesically equivalent metrics is Lp,q or Seifertmanifold with zero Euler number.)Assume dim(M) = 3


Case 1: There exists a point of the manifold such that that thepolynomial det(g − λg) has 3 different roots. Then, the geodesicflow of g is Liouville-integrable.


Case 1: There exists a point of the manifold such that that thepolynomial det(g − λg) has 3 different roots. Then, the geodesicflow of g is Liouville-integrable.Theorem ( Kruglikov, Matveev 2006): Then, the topologicalentropy of g vanishes.


Case 1: There exists a point of the manifold such that that thepolynomial det(g − λg) has 3 different roots. Then, the geodesicflow of g is Liouville-integrable.Theorem ( Kruglikov, Matveev 2006): Then, the topologicalentropy of g vanishes.(And therefore modulo the Poincare conjecture the manifold canbe covered by S3, S2 × S1 or by S1 × S1 × S1.)


Case 1: There exists a point of the manifold such that that thepolynomial det(g − λg) has 3 different roots. Then, the geodesicflow of g is Liouville-integrable.Theorem ( Kruglikov, Matveev 2006): Then, the topologicalentropy of g vanishes.(And therefore modulo the Poincare conjecture the manifold canbe covered by S3, S2 × S1 or by S1 × S1 × S1.)


Case 1: There exists a point of the manifold such that that thepolynomial det(g − λg) has 3 different roots. Then, the geodesicflow of g is Liouville-integrable.Theorem ( Kruglikov, Matveev 2006): Then, the topologicalentropy of g vanishes.(And therefore modulo the Poincare conjecture the manifold canbe covered by S3, S2 × S1 or by S1 × S1 × S1.)Case 2: At every point the number of roots of the polynomial is≤ 2.Then precisely the same trick as in dimension 2 works.


Case 1: There exists a point of the manifold such that that thepolynomial det(g − λg) has 3 different roots. Then, the geodesicflow of g is Liouville-integrable.Theorem ( Kruglikov, Matveev 2006): Then, the topologicalentropy of g vanishes.(And therefore modulo the Poincare conjecture the manifold canbe covered by S3, S2 × S1 or by S1 × S1 × S1.)Case 2: At every point the number of roots of the polynomial is≤ 2.Then precisely the same trick as in dimension 2 works.

Combining the methods from integrable systems anddifferential geometry: Lie problem


Lie 1882: Problem II: Man soll die Form des Bo-genelementes einer jeden Flache bestimmen,deren geodatische Kurven mehrere infinitesimaleTransformationen gestatten


Lie 1882: Problem II: Man soll die Form des Bo-genelementes einer jeden Flache bestimmen,deren geodatische Kurven mehrere infinitesimaleTransformationen gestattenEnglish translation: Describe all 2 dim metrics admitting at leasttwo projective vector fields


Lie 1882: Problem II: Man soll die Form des Bo-genelementes einer jeden Flache bestimmen,deren geodatische Kurven mehrere infinitesimaleTransformationen gestattenEnglish translation: Describe all 2 dim metrics admitting at leasttwo projective vector fieldsRepeating Def: A vector field is projective (w.r.t. a metric), if itsflow takes unparameterized geodesics to geodesics.

Theorem (Bryant, Manno, M∼ 2007) If a two-dimensionalmetric g of nonconstant curvature has at least 2 projective vectorfields such that they are linear independent at the point p, thenthere exist coordinates x , y in a neighborhood of p such that themetrics are as follows.

Theorem (Bryant, Manno, M∼ 2007) If a two-dimensionalmetric g of nonconstant curvature has at least 2 projective vectorfields such that they are linear independent at the point p, thenthere exist coordinates x , y in a neighborhood of p such that themetrics are as follows.

1. ǫ1e(b+2) xdx2 + ǫ2beb xdy2, where

b ∈ R \ {−2, 0, 1} and ǫi ∈ {−1, 1}

2. a(

ǫ1e(b+2) xdx2

(eb x+ǫ2)2+ eb xdy2

eb x+ǫ2

)

, where a ∈ R \ {0},

b ∈ R \ {−2, 0, 1} and ǫi ∈ {−1, 1}

3. a(

e2 xdx2

x2 + ǫdy2

x

)

, where a ∈ R \ {0}, and ǫ ∈ {−1, 1}

4. ǫ1e3xdx2 + ǫ2e

xdy2, where ǫi ∈ {−1, 1},

5. a(

e3xdx2

(ex+ǫ2)2+ ǫ1e

xdy2

(ex+ǫ2)

)

, where a ∈ R \ {0}, ǫi ∈ {−1, 1},

6. a(

dx2

(cx+2x2+ǫ2)2x+ ǫ1

xdy2

cx+2x2+ǫ2

)

, where a > 0, ǫi ∈ {−1, 1},

c ∈ R.

Schouten 1924: List all complete metrics admittingcomplete projective vector field


I proved Lichnerowicz-Obata-Solodovnikov Conjecture (50th):


I proved Lichnerowicz-Obata-Solodovnikov Conjecture (50th): Let acomplete Riemannian manifold (of dim ≥ 2) admit a complete projectivevector field.


I proved Lichnerowicz-Obata-Solodovnikov Conjecture (50th): Let acomplete Riemannian manifold (of dim ≥ 2) admit a complete projectivevector field. Then, the manifold is covered by the round sphere, or thevector field is affine.





It is hard to relax the assumptions







History of L-O-S conjecture:









France(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 conjecture










Solodovnikov (1956)proved the conjectureassuming that all ob-jects are real analytic










Solodovnikov (1956)proved the conjectureassuming that all ob-jects are real analytic










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










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

Pseudo-Riemannian case was considered: Venzi 85, Barnes 93, Hall 95.

Geometric theory of PDE: Solution of Lichnerowich/Lie



(Lichnerowicz Conjecture: Among closed Riemannian manifolds, onlythe round sphere admits essential projective vector fields)


(Lichnerowicz Conjecture: Among closed Riemannian manifolds, onlythe round sphere admits essential projective vector fields)


(Lichnerowicz Conjecture: Among closed Riemannian manifolds, onlythe round sphere admits essential projective vector fields)(Lie Problem: Describe all 2-dim metrics with two projective vector

fields)



fields)

We reformulate it on the language of PDE.



fields)

We reformulate it on the language of PDE.Folklor (Levi-Civita 1896): Two symmetric affine connections Γ and Γhave the same (unparametrized) geodesics (=belong to the sameprojective class),



fields)

We reformulate it on the language of PDE.Folklor (Levi-Civita 1896): Two symmetric affine connections Γ and Γhave the same (unparametrized) geodesics (=belong to the sameprojective class), iff for a 1-form Υ



fields)

We reformulate it on the language of PDE.Folklor (Levi-Civita 1896): Two symmetric affine connections Γ and Γhave the same (unparametrized) geodesics (=belong to the sameprojective class), iff for a 1-form ΥΓi

jk = Γijk + Υjδ

ik + Υkδ

ij



fields)


jk = Γijk + Υjδ

ik + Υkδ

ij



fields)


jk = Γijk + Υjδ

ik + Υkδ

ij

(Alternatively:



fields)


jk = Γijk + Υjδ

ik + Υkδ

ij

(Alternatively: Tracefreepartof(Γ − Γ) = 0). )



fields)


jk = Γijk + Υjδ

ik + Υkδ

ij


Then, on the level of equations



fields)


jk = Γijk + Υjδ

ik + Υkδ

ij



All metrics geodesically equivalent to a connection Γ are solutions of thesystem



fields)


jk = Γijk + Υjδ

ik + Υkδ

ij



All metrics geodesically equivalent to a connection Γ are solutions of thesystem[

∂∂k

gij − giαΓαkj − gjαΓα

ki

]

− Υigkj − Υjgki = 0



fields)


jk = Γijk + Υjδ

ik + Υkδ

ij




∂∂k


ki

]


on unknowns gij and Υi .



fields)


jk = Γijk + Υjδ

ik + Υkδ

ij




∂∂k


ki

]


on unknowns gij and Υi . The system is nonlinear



fields)


jk = Γijk + Υjδ

ik + Υkδ

ij




∂∂k


ki

]


on unknowns gij and Υi . The system is nonlinear and

contains artificial unknown Υ

Trick: One can exclude Υ and linearize the system:


Dim 2: R. Liouville 1889.


Dim 2: R. Liouville 1889. : Liouville regarded this linearity as aremarkably lucky circumstance


Dim 2: R. Liouville 1889. : Liouville regarded this linearity as aremarkably lucky circumstance


Dim 2: R. Liouville 1889. : Liouville regarded this linearity as aremarkably lucky circumstanceDim n: Sinjukov 1962/


Dim 2: R. Liouville 1889. : Liouville regarded this linearity as aremarkably lucky circumstanceDim n: Sinjukov 1962/ Mikes, Berezovski 1989 /


Dim 2: R. Liouville 1889. : Liouville regarded this linearity as aremarkably lucky circumstanceDim n: Sinjukov 1962/ Mikes, Berezovski 1989 / Bolsinov, M∼ 2003.


Dim 2: R. Liouville 1889. : Liouville regarded this linearity as aremarkably lucky circumstanceDim n: Sinjukov 1962/ Mikes, Berezovski 1989 / Bolsinov, M∼ 2003.Theorem (Eastwood, M∼ 2007) g is geodesically equivalent to aconnection Γi

jk iff σab := g ab · det(g)1/(n+1) is a solution of




Tracefreepartof(∇aσ

bc)

= 0.





bc)

= 0.

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

2/(n+1)M.





bc)

= 0.


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





bc)

= 0.



∇aσbc =

∂

∂xaσbc + Γb

adσdc + Γcdaσ

bd

︸︷︷︸


−2

n + 1Γd

da σbc

︸︷︷︸


Advantages:





bc)

= 0.



∇aσbc =

∂

∂xaσbc + Γb

adσdc + Γcdaσ

bd

︸︷︷︸


−2

n + 1Γd

da σbc

︸︷︷︸


Advantages:

1. It is a linear PDE-system of the first order.





bc)

= 0.



∇aσbc =

∂

∂xaσbc + Γb

adσdc + Γcdaσ

bd

︸︷︷︸


−2

n + 1Γd

da σbc

︸︷︷︸


Advantages:


2. The system does not depend on the choice of Γ within a projectiveclass.





bc)

= 0.



∇aσbc =

∂

∂xaσbc + Γb

adσdc + Γcdaσ

bd

︸︷︷︸


−2

n + 1Γd

da σbc

︸︷︷︸


Advantages:


2. The system does not depend on the choice of Γ within a projectiveclass. (short tensor calculations)

How to use system: dim 2: preliminaries


Def: Projective connection of an affine connection Γ is

yxx = −Γ211 + (Γ1

11 − 2Γ212)yx

+ (−Γ222 + 2Γ1

12)(yx)2 + Γ1

22(yx)3



yxx = −Γ211 + (Γ1

11 − 2Γ212)yx

+ (−Γ222 + 2Γ1

12)(yx)2 + Γ1

22(yx)3

Beltrami 1865: For every solution y(x) the curve (x , y(x)) is are-parametrized geodesic.



yxx = −Γ211 + (Γ1

11 − 2Γ212)yx

+ (−Γ222 + 2Γ1

12)(yx)2 + Γ1

22(yx)3

Beltrami 1865: For every solution y(x) the curve (x , y(x)) is are-parametrized geodesic.Corollary If g has two projective vector fields, then its projectiveconnection has two point-symmetries.



yxx = −Γ211 + (Γ1

11 − 2Γ212)yx

+ (−Γ222 + 2Γ1

12)(yx)2 + Γ1

22(yx)3

Beltrami 1865: For every solution y(x) the curve (x , y(x)) is are-parametrized geodesic.Corollary If g has two projective vector fields, then its projectiveconnection has two point-symmetries.



yxx = −Γ211 + (Γ1

11 − 2Γ212)yx

+ (−Γ222 + 2Γ1

12)(yx)2 + Γ1

22(yx)3

Beltrami 1865: For every solution y(x) the curve (x , y(x)) is are-parametrized geodesic.Corollary If g has two projective vector fields, then its projectiveconnection has two point-symmetries.It is easy to describe such projective connections (Lie, Cartan, Tresse):



yxx = −Γ211 + (Γ1

11 − 2Γ212)yx

+ (−Γ222 + 2Γ1

12)(yx)2 + Γ1

22(yx)3

Beltrami 1865: For every solution y(x) the curve (x , y(x)) is are-parametrized geodesic.Corollary If g has two projective vector fields, then its projectiveconnection has two point-symmetries.It is easy to describe such projective connections (Lie, Cartan, Tresse):in a certain coordinate system they are as follows: (A,B,C ,D ∈ R )yxx = exA + Byx + Ce−x(yx)

2 + De−2x(yx)3.



yxx = −Γ211 + (Γ1

11 − 2Γ212)yx

+ (−Γ222 + 2Γ1

12)(yx)2 + Γ1

22(yx)3

Beltrami 1865: For every solution y(x) the curve (x , y(x)) is are-parametrized geodesic.Corollary If g has two projective vector fields, then its projectiveconnection has two point-symmetries.It is easy to describe such projective connections (Lie, Cartan, Tresse):in a certain coordinate system they are as follows: (A,B,C ,D ∈ R )yxx = exA + Byx + Ce−x(yx)

2 + De−2x(yx)3.

Lie Problem is equivalent to “Which of them come from a metric”

Theorem (R. Liouville 1889 – Eastwood/ M∼ 2007–Bryant/Manno/M∼ 2007)

Theorem (R. Liouville 1889 – Eastwood/ M∼ 2007–Bryant/Manno/M∼ 2007) A 2-dimensional metric g has a givenprojective connection

yxx = K0 + K1yx + K2(yx)2 + K3(yx)

3


yxx = K0 + K1yx + K2(yx)2 + K3(yx)

3

iff the entries of the matrix adef= (det(g))

−2/3g satisfy a system of 4

linear PDE of first order.


yxx = K0 + K1yx + K2(yx)2 + K3(yx)

3




∂a11

∂x+ 2K0a12 − 2/3K1a11 = 0

2 ∂a12

∂x+ ∂a11

∂y+ 2K0a22 + 2/3K1a12 − 4/3K2a11 = 0

∂a22

∂x+ 2 ∂a12

∂y+ 4/3K1a22 − 2/3K2a12 − 2K3a11 = 0

∂a22

∂y+ 2/3K2a22 − 2K3a12 = 0


yxx = K0 + K1yx + K2(yx)2 + K3(yx)

3




∂a11

∂x+ 2K0a12 − 2/3K1a11 = 0

2 ∂a12

∂x+ ∂a11

∂y+ 2K0a22 + 2/3K1a12 − 4/3K2a11 = 0

∂a22

∂x+ 2 ∂a12

∂y+ 4/3K1a22 − 2/3K2a12 − 2K3a11 = 0

∂a22

∂y+ 2/3K2a22 − 2K3a12 = 0

Thus, the projective connection came from a metric iff this systemof PDE has a nontrivial solution.


yxx = K0 + K1yx + K2(yx)2 + K3(yx)

3




∂a11

∂x+ 2K0a12 − 2/3K1a11 = 0

2 ∂a12

∂x+ ∂a11

∂y+ 2K0a22 + 2/3K1a12 − 4/3K2a11 = 0

∂a22

∂x+ 2 ∂a12

∂y+ 4/3K1a22 − 2/3K2a12 − 2K3a11 = 0

∂a22

∂y+ 2/3K2a22 − 2K3a12 = 0


To solve the Lie problem = to determine constants A,B,C ,D


yxx = K0 + K1yx + K2(yx)2 + K3(yx)

3




∂a11

∂x+ 2K0a12 − 2/3K1a11 = 0

2 ∂a12

∂x+ ∂a11

∂y+ 2K0a22 + 2/3K1a12 − 4/3K2a11 = 0

∂a22

∂x+ 2 ∂a12

∂y+ 4/3K1a22 − 2/3K2a12 − 2K3a11 = 0

∂a22

∂y+ 2/3K2a22 − 2K3a12 = 0


To solve the Lie problem = to determine constants A,B,C ,D suchthat the system above withK0 = exA, K1 = B, K2 = Ce−x , K3 = De−2x


yxx = K0 + K1yx + K2(yx)2 + K3(yx)

3




∂a11

∂x+ 2K0a12 − 2/3K1a11 = 0

2 ∂a12

∂x+ ∂a11

∂y+ 2K0a22 + 2/3K1a12 − 4/3K2a11 = 0

∂a22

∂x+ 2 ∂a12

∂y+ 4/3K1a22 − 2/3K2a12 − 2K3a11 = 0

∂a22

∂y+ 2/3K2a22 − 2K3a12 = 0



has a nontrivial solution


yxx = K0 + K1yx + K2(yx)2 + K3(yx)

3




∂a11

∂x+ 2K0a12 − 2/3K1a11 = 0

2 ∂a12

∂x+ ∂a11

∂y+ 2K0a22 + 2/3K1a12 − 4/3K2a11 = 0

∂a22

∂x+ 2 ∂a12

∂y+ 4/3K1a22 − 2/3K2a12 − 2K3a11 = 0

∂a22

∂y+ 2/3K2a22 − 2K3a12 = 0




This is an algorithmically-solvable problem


yxx = K0 + K1yx + K2(yx)2 + K3(yx)

3




∂a11

∂x+ 2K0a12 − 2/3K1a11 = 0

2 ∂a12

∂x+ ∂a11

∂y+ 2K0a22 + 2/3K1a12 − 4/3K2a11 = 0

∂a22

∂x+ 2 ∂a12

∂y+ 4/3K1a22 − 2/3K2a12 − 2K3a11 = 0

∂a22

∂y+ 2/3K2a22 − 2K3a12 = 0




This is an algorithmically-solvable problem(if we can differentiate, multiply and add).

Standard algorithm to understand the dimension of the space ofsolutions

Standard algorithm to understand the dimension of the space ofsolutionsDifferentiate every equation of the above system w.r.t. xx , xy , yy

Standard algorithm to understand the dimension of the space ofsolutionsDifferentiate every equation of the above system w.r.t. xx , xy , yy andsolve the obtained equations w.r.t. third derivatives.


(∗)


(∗)

∂3a11

∂x3 = f11111(x , y , aαβ ,∂aαβ

∂xγ,

∂2aαβ

∂2xγ∂xδ)

...∂3a22

∂y3 = f22222(x , y , aαβ ,∂aαβ

∂xγ,

∂2aαβ

∂xγ∂xδ)


(∗)

∂3a11

∂x3 = f11111(x , y , aαβ ,∂aαβ

∂xγ,

∂2aαβ

∂2xγ∂xδ)

...∂3a22

∂y3 = f22222(x , y , aαβ ,∂aαβ

∂xγ,

∂2aαβ

∂xγ∂xδ)

Now, let us produce new linear PDE’s of the second order


(∗)

∂3a11

∂x3 = f11111(x , y , aαβ ,∂aαβ

∂xγ,

∂2aαβ

∂2xγ∂xδ)

...∂3a22

∂y3 = f22222(x , y , aαβ ,∂aαβ

∂xγ,

∂2aαβ

∂xγ∂xδ)


Step 1: Take∂fijkrm

∂xs−

∂fijkrs∂xm


(∗)

∂3a11

∂x3 = f11111(x , y , aαβ ,∂aαβ

∂xγ,

∂2aαβ

∂2xγ∂xδ)

...∂3a22

∂y3 = f22222(x , y , aαβ ,∂aαβ

∂xγ,

∂2aαβ

∂xγ∂xδ)



∂xs−

∂fijkrs∂xm

and substitute (∗) inside.


(∗)

∂3a11

∂x3 = f11111(x , y , aαβ ,∂aαβ

∂xγ,

∂2aαβ

∂2xγ∂xδ)

...∂3a22

∂y3 = f22222(x , y , aαβ ,∂aαβ

∂xγ,

∂2aαβ

∂xγ∂xδ)



∂xs−

∂fijkrs∂xm


Step 2: Differentiate the obtained equations by xα and substitute (∗)inside.


(∗)

∂3a11

∂x3 = f11111(x , y , aαβ ,∂aαβ

∂xγ,

∂2aαβ

∂2xγ∂xδ)

...∂3a22

∂y3 = f22222(x , y , aαβ ,∂aαβ

∂xγ,

∂2aαβ

∂xγ∂xδ)



∂xs−

∂fijkrs∂xm


Step 2: Differentiate the obtained equations by xα and substitute (∗)inside.Step 3: Differentiate the obtained equations by xα and substitute (∗)inside.


(∗)

∂3a11

∂x3 = f11111(x , y , aαβ ,∂aαβ

∂xγ,

∂2aαβ

∂2xγ∂xδ)

...∂3a22

∂y3 = f22222(x , y , aαβ ,∂aαβ

∂xγ,

∂2aαβ

∂xγ∂xδ)



∂xs−

∂fijkrs∂xm


Step 2: Differentiate the obtained equations by xα and substitute (∗)inside.Step 3: Differentiate the obtained equations by xα and substitute (∗)inside.Step 4: Differentiate the obtained equations by xα and substitute (∗)inside.


(∗)

∂3a11

∂x3 = f11111(x , y , aαβ ,∂aαβ

∂xγ,

∂2aαβ

∂2xγ∂xδ)

...∂3a22

∂y3 = f22222(x , y , aαβ ,∂aαβ

∂xγ,

∂2aαβ

∂xγ∂xδ)



∂xs−

∂fijkrs∂xm



Step 100: Differentiate the obtained equations by xα and substitute (∗)

inside.


(∗)

∂3a11

∂x3 = f11111(x , y , aαβ ,∂aαβ

∂xγ,

∂2aαβ

∂2xγ∂xδ)

...∂3a22

∂y3 = f22222(x , y , aαβ ,∂aαβ

∂xγ,

∂2aαβ

∂xγ∂xδ)



∂xs−

∂fijkrs∂xm



Step 100: Differentiate the obtained equations by xα and substitute (∗)

inside.

At a certain point all new equations we obtain are linearcombinations of the previous equations.

At a certain point all new equations we obtain are linearcombinations of the previous equations.Then, 18 minus the rank of the systems of equation we obtained isthe dimension of the space of solutions of (∗), i.e. is the dimensionof the space of the metrics corresponding to the above projectiveconnection.

At a certain point all new equations we obtain are linearcombinations of the previous equations.Then, 18 minus the rank of the systems of equation we obtained isthe dimension of the space of solutions of (∗), i.e. is the dimensionof the space of the metrics corresponding to the above projectiveconnection.Applying the above procedure to the projective connections

admitting 2 point symmetries, we solve the Lie 2nd problem:



Theorem (Bryant, Manno, M∼).yxx = exA + Byx + Ce−x(yx)

2 + De−2x(yx)3 comes from a metric

iff A = C = 0 (may be after a coordinate change)



Theorem (Bryant, Manno, M∼).yxx = exA + Byx + Ce−x(yx)

2 + De−2x(yx)3 comes from a metric

iff A = C = 0 (may be after a coordinate change)

Of cause, we also have a hand-written proof.

Lichnerowicz problem

Denote by σ the space of solutions.


Denote by σ the space of solutions.Assume σ is two-dimenional.


Denote by σ the space of solutions.Assume σ is two-dimenional.Every projective vector field v does not change the equations.


Denote by σ the space of solutions.Assume σ is two-dimenional.Every projective vector field v does not change the equations.Then,


Denote by σ the space of solutions.Assume σ is two-dimenional.Every projective vector field v does not change the equations.Then, for every solution σ ∈ σ,


Denote by σ the space of solutions.Assume σ is two-dimenional.Every projective vector field v does not change the equations.Then, for every solution σ ∈ σ, then Lvσ is also a soluiton. Thus, Lv isa linear mapping :σ → σ.


Denote by σ the space of solutions.Assume σ is two-dimenional.Every projective vector field v does not change the equations.Then, for every solution σ ∈ σ, then Lvσ is also a soluiton. Thus, Lv isa linear mapping :σ → σ.Then, for every basis σ1, σ2 ∈ σ

Lv

�σ1σ2

�



Lv

�σ1σ2

�= A

�σ1σ2

�,



Lv

�σ1σ2

�= A

�σ1σ2

�,

where A is a 2 × 2 matrix.



Lv

�σ1σ2

�= A

�σ1σ2

�,

where A is a 2 × 2 matrix. Then, for the appropiate choice of the basisσ1, σ2 we get



Lv

�σ1σ2

�= A

�σ1σ2

�,

where A is a 2 × 2 matrix. Then, for the appropiate choice of the basisσ1, σ2 we getLv

�σ1σ2

�=



Lv

�σ1σ2

�= A

�σ1σ2

�,


�σ1σ2

�=

�α ∗

0 β

��σ1σ2

�, or



Lv

�σ1σ2

�= A

�σ1σ2

�,


�σ1σ2

�=

�α ∗

0 β

��σ1σ2

�, or Lv

�σ1σ2

�=

�α β

−β α

��σ1σ2

�



Lv

�σ1σ2

�= A

�σ1σ2

�,


�σ1σ2

�=

�α ∗

0 β

��σ1σ2

�, or Lv

�σ1σ2

�=

�α β

−β α

��σ1σ2

�depending on the eigenvalues of A.

In the left case



Lv

�σ1σ2

�= A

�σ1σ2

�,


�σ1σ2

�=

�α ∗

0 β

��σ1σ2

�, or Lv

�σ1σ2

�=

�α β

−β α

��σ1σ2


In the left case there exists a solution s.t. Lvσ = ασ



Lv

�σ1σ2

�= A

�σ1σ2

�,


�σ1σ2

�=

�α ∗

0 β

��σ1σ2

�, or Lv

�σ1σ2

�=

�α β

−β α

��σ1σ2


In the left case there exists a solution s.t. Lvσ = ασ implying v is

homothety vector for g



Lv

�σ1σ2

�= A

�σ1σ2

�,


�σ1σ2

�=

�α ∗

0 β

��σ1σ2

�, or Lv

�σ1σ2

�=

�α β

−β α

��σ1σ2



homothety vector for g which is impossible for closed manifolds.



Lv

�σ1σ2

�= A

�σ1σ2

�,


�σ1σ2

�=

�α ∗

0 β

��σ1σ2

�, or Lv

�σ1σ2

�=

�α β

−β α

��σ1σ2



homothety vector for g which is impossible for closed manifolds. In the

second case



Lv

�σ1σ2

�= A

�σ1σ2

�,


�σ1σ2

�=

�α ∗

0 β

��σ1σ2

�, or Lv

�σ1σ2

�=

�α β

−β α

��σ1σ2




second case the determinant of σ vanishes at some point



Lv

�σ1σ2

�= A

�σ1σ2

�,


�σ1σ2

�=

�α ∗

0 β

��σ1σ2

�, or Lv

�σ1σ2

�=

�α β

−β α

��σ1σ2




second case the determinant of σ vanishes at some point (because

cos(t) vanishes for some t)



Lv

�σ1σ2

�= A

�σ1σ2

�,


�σ1σ2

�=

�α ∗

0 β

��σ1σ2

�, or Lv

�σ1σ2

�=

�α β

−β α

��σ1σ2





cos(t) vanishes for some t) implying that g = σ · det(σ) vanishes

somewhere.



Lv

�σ1σ2

�= A

�σ1σ2

�,


�σ1σ2

�=

�α ∗

0 β

��σ1σ2

�, or Lv

�σ1σ2

�=

�α β

−β α

��σ1σ2





cos(t) vanishes for some t) implying that g = σ · det(σ) vanishes

somewhere. Lichnerowich conjecture is proved under assumption that

the space of solutions is two-dimenional

An attempt to explain why the system is linear. (Worksonly in dim 2)

We know (first part of the talk:)


We know (first part of the talk:) if g and g have the same

geodesics, then I0(ξ) :=(

det(g)det(g)

) 23g(ξ, ξ)




det(g)det(g)

) 23g(ξ, ξ) is an integral of the

geodesic flow of g , i.e.




det(g)det(g)



{(

det(g)det(g)

)23

g , g

}

g

= 0, (∗)




det(g)det(g)



{(

det(g)det(g)

)23

g , g

}

g

= 0, (∗)

where { , }g is the Poisson bracket transplanted from T ∗M to TMby the bundle isomorphism ♭g : TM → T ∗M.




det(g)det(g)



{(

det(g)det(g)

)23

g , g

}

g

= 0, (∗)

where { , }g is the Poisson bracket transplanted from T ∗M to TMby the bundle isomorphism ♭g : TM → T ∗M. If g is known,




det(g)det(g)



{(

det(g)det(g)

)23

g , g

}

g

= 0, (∗)


this equations is visibly linear in a = g/ det(g)2/3




det(g)det(g)



{(

det(g)det(g)

)23

g , g

}

g

= 0, (∗)


this equations is visibly linear in a = g/ det(g)2/3 . Since in dim 2the mapping A 7→ comatrix(A) = A−1 · det(A) is linear




det(g)det(g)



{(

det(g)det(g)

)23

g , g

}

g

= 0, (∗)


this equations is visibly linear in a = g/ det(g)2/3 . Since in dim 2the mapping A 7→ comatrix(A) = A−1 · det(A) is linear , themapping g−1 · det(g)1/3 7→ g/ det(g)2/3 is a linear isomorphism




det(g)det(g)



{(

det(g)det(g)

)23

g , g

}

g

= 0, (∗)


this equations is visibly linear in a = g/ det(g)2/3 . Since in dim 2the mapping A 7→ comatrix(A) = A−1 · det(A) is linear , themapping g−1 · det(g)1/3 7→ g/ det(g)2/3 is a linear isomorphism ,and the equations (∗) can be viewed as linear PDE in in thecomponents of σ := g−1 · det(g)1/3.

What happens if the space of solutions is more thantwo-dimensional?


Theorem (Matveev, 2003) Assume Γ is the Levi-Civita connection of

a Riemannian metric on a closed manifold.



a Riemannian metric on a closed manifold. If dim(σ)> 3, then g hasconstant curvature



a Riemannian metric on a closed manifold. If dim(σ)> 3, then g hasconstant curvatureExplanation:



a Riemannian metric on a closed manifold. If dim(σ)> 3, then g hasconstant curvatureExplanation: The above linear system of PDE can be prolonged:

∇aσbc = δa

bµc + δacµb

∇aµb = δa

bρ − 1nPacσ

bc + 1nWac

bdσcd

∇aρ = − 2nPabµ

b + 4nYabcσ

bc

(Mikes 1998

Eastwood, M∼ 2007

)




∇aσbc = δa

bµc + δacµb

∇aµb = δa

bρ − 1nPacσ

bc + 1nWac

bdσcd


b + 4nYabcσ

bc

(Mikes 1998

Eastwood, M∼ 2007

)

where P is the symmeterized Ricci-tensor, Y the Cotton-York-Tensor andWab

cd the projective Weyl-Tensor for the connection Γ.




∇aσbc = δa

bµc + δacµb

∇aµb = δa

bρ − 1nPacσ

bc + 1nWac

bdσcd


b + 4nYabcσ

bc

(Mikes 1998

Eastwood, M∼ 2007

)



This is a linear system of PDE of the first order on the unknownfunctions σbc , µb , ρ.




∇aσbc = δa

bµc + δacµb

∇aµb = δa

bρ − 1nPacσ

bc + 1nWac

bdσcd


b + 4nYabcσ

bc

(Mikes 1998

Eastwood, M∼ 2007

)



This is a linear system of PDE of the first order on the unknownfunctions σbc , µb , ρ. Moreover,

1. All derivatives of unknowns are expressed as functions of unknowns,i.e. it is a connection.




∇aσbc = δa

bµc + δacµb

∇aµb = δa

bρ − 1nPacσ

bc + 1nWac

bdσcd


b + 4nYabcσ

bc

(Mikes 1998

Eastwood, M∼ 2007

)



This is a linear system of PDE of the first order on the unknownfunctions σbc , µb , ρ. Moreover,

1. All derivatives of unknowns are expressed as functions of unknowns,i.e. it is a connection.

2. One can understood this connection as a connection on theprojective tractor bundle E (BC) = E (bc)(−2) + Eb(−2) + E(−2)

Its curvature (integrability conditions) can be explicitely calculated

Its curvature (integrability conditions) can be explicitely calculated(partially done by Solodovnikov, 1956) .

Its curvature (integrability conditions) can be explicitely calculated(partially done by Solodovnikov, 1956) . The solutions of the systems areparallel sections of the connection,

Its curvature (integrability conditions) can be explicitely calculated(partially done by Solodovnikov, 1956) . The solutions of the systems areparallel sections of the connection, and the existence of n−dimensionalspace of is just the condition that the holonomy fix three-dimensionaldistribution.

Its curvature (integrability conditions) can be explicitely calculated(partially done by Solodovnikov, 1956) . The solutions of the systems areparallel sections of the connection, and the existence of n−dimensionalspace of is just the condition that the holonomy fix three-dimensionaldistribution. Thus condition allows to canonicaly construct smoothfunctions which


◮ vanish identically iff the metric has constant curvature (localcondition)



◮ explode if the manifold is closed and if they are not zero



◮ explode if the manifold is closed and if they are not zero

Open questions and possible collaborations


◮ To solve the Lichnerowich conjecture in thepseudo-Riemannian case.


◮ To solve the Lichnerowich conjecture in thepseudo-Riemannian case. Joint project with Bolsinov andKiosak

◮ One can try to apply Topalov-Sinjukov hierarhy



◮ One can try to apply Topalov-Sinjukov hierarhy◮ as a source of integrable systems



◮ One can try to apply Topalov-Sinjukov hierarhy◮ as a source of integrable systems◮ as test systems for different ways of quantizing




◮ one can try to combine geodesic equivalence with additionaltensor assumptions





◮ the metric is Einstein (Joint project with Kiosak)





◮ the metric is Einstein (Joint project with Kiosak)◮ the metric is Kahler











◮ To generalize for pseudo-Riemannian case






◮ To generalize for pseudo-Riemannian case (local descriptionin dim 2 – Bolsinov Pucacco M∼ 2008)













Thanks a lot!!!

Documents

Vladimir Matveev Jena (Germany) › ~matveev › Datei › presentation_rome.pdfTwo separately developed theories, theory of geodesically equivalent metrics and theory of quadratically