History Projective differential geometry is a very classical topic (at least under the assumption that the connections Γ and ¯ Γ are Levi-Civita connnections): the first examples are due to Lagrange 1779 and many questions were posed and solved (and sometimes remain unsolved) by classics of differential geometry and mechanics: Beltrami, Dini, Levi-Civita, Painleve, Weyl,... The people in the h − projective geometry, as a rule, have projective-geometry as a background: there is one recent exception from this 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 define h -projectively equivalent metrics, because in the K¨ ahler situation projective equivalence is not interesting. Most questions studied in the projective geometry can be generalised to the h -projective setting – and these is what most people before approx. 2003 did with the hope that they can also generalize the proofs.
History Projective differential geometry is a very classical topic (at least under the assumption that the connections Γ and ¯ Γ are Levi-Civita connnections): the first examples are due to Lagrange 1779 and many questions were posed and solved (and sometimes remain unsolved) by classics of differential geometry and mechanics: Beltrami, Dini, Levi-Civita, Painleve, Weyl,... The people in the h − projective geometry, as a rule, have projective-geometry as a background: there is one recent exception from this 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 define h -projectively equivalent metrics, because in the K¨ ahler situation projective equivalence is not interesting. Most questions studied in the projective geometry can be generalised to the 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 reinvented h -projective geometry in completely different terms: Kiyohara - Topalov: What they called “typ A K¨ ahler-Liouville systems” is a special case of h -projectively equivalent metrics. Apostolov-Calderbank-Gauduchon: What they called “Hamiltonian 2-forms” is precisely the same as h -projectively equivalent metrics.
Around 2003 two strong teams reinvented h -projective geometry in completely different terms: Kiyohara - Topalov: What they called “typ A K¨ ahler-Liouville systems” is a special case of h -projectively equivalent metrics. Apostolov-Calderbank-Gauduchon: What they called “Hamiltonian 2-forms” is precisely the same as h -projectively equivalent metrics. These groups brought new technigue in the subject: integrable systems technique, symplectic and K¨ ahler geomety technique, and parabolic geometry technique. These new techniques together can effectively help to solve the problems stated by classics – I will show two examples.
Symmetries of the projective and h -projective structures.
Symmetries of the projective and h -projective structures. Def. A vector field is a symmetry of a projective structure, if it sends geodesics to geodesics.
Symmetries of the projective and h -projective structures. Def. A vector field is a symmetry of a projective structure, if it sends geodesics to geodesics. Def. A vector field is a symmetry of a h -projective structure, if it sends h − planar curves to h -planar curves.
Symmetries of the projective and h -projective structures. Def. A vector field is a symmetry of a projective structure, if it sends geodesics to geodesics. Def. A vector field is a symmetry of a h -projective structure, if it sends h − planar curves to h -planar curves. ∂ Easy Theorem 1. A vector field ∂ x 1 is a symmetry of a projective structure [Γ], iff jk ( x 1 , ..., x n ) = ˇ jk ( x 2 , ..., x n j φ k ( x 1 , ..., x n ) + δ i k φ j ( x 1 , ..., x n ) . Γ i Γ i ) + δ i � �� � no x 1 − coord.
Symmetries of the projective and h -projective structures. Def. A vector field is a symmetry of a projective structure, if it sends geodesics to geodesics. Def. A vector field is a symmetry of a h -projective structure, if it sends h − planar curves to h -planar curves. ∂ Easy Theorem 1. A vector field ∂ x 1 is a symmetry of a projective structure [Γ], iff jk ( x 1 , ..., x n ) = ˇ jk ( x 2 , ..., x n j φ k ( x 1 , ..., x n ) + δ i k φ j ( x 1 , ..., x n ) . Γ i Γ i ) + δ i � �� � no x 1 − coord. ∂ Easy Theorem 2. A vector field ∂ x 1 is a symmetry of a h -projective structure [Γ], iff ˇ jk ( x 2 , ..., x n Γ i jk ( x 1 , ..., x n ) = Γ i ) + δ i j φ k ( x 1 , ..., x n ) + δ i k φ j ( x 1 , ..., x n ) � �� � no x 1 − coord. − J i j φ a ( x 1 , ..., x n ) J a k − J i k φ a ( x 1 , ..., x n ) J a j . Thus, there is almost no sense to study a symmetry of projective or h − projective structures.
I will discuss “metric” h-projective geometry I assume that the projective (or h -projective) structure contains the Levi-Civita connection of a (pseudo-)Riemannian metric.
I will discuss “metric” h-projective geometry I assume that the projective (or h -projective) structure contains the Levi-Civita connection of a (pseudo-)Riemannian metric. That means, I am speaking not about ( h -)projectively equivalent connections, but about ( h -)projectively equivalent metrics.
I will discuss “metric” h-projective geometry I assume that the projective (or h -projective) structure contains the Levi-Civita connection of a (pseudo-)Riemannian metric. That means, I am speaking not about ( h -)projectively equivalent connections, but about ( h -)projectively equivalent metrics. It appeared though that this is convenient to fix a connection and then 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 to Liouville 1889, Sinjukov 1961, Bolsinov-Matveev 2003) The Levi-Civita connection of g lies in a projective class of a connection Γ i jk if and only if σ ab := g ab · det( g ) 1 / ( n +1) is a solution of � ∇ a σ bc � � � 1 ∇ i σ ib δ c a + ∇ i σ ic δ b = 0 . ( ∗ ) − a n +1
Theorem (Eastwood-Matveev 2006; special cases known to Liouville 1889, Sinjukov 1961, Bolsinov-Matveev 2003) The Levi-Civita connection of g lies in a projective class of a connection Γ i jk if and only if σ ab := g ab · det( g ) 1 / ( n +1) is a solution of � ∇ a σ bc � � � 1 ∇ i σ ib δ c a + ∇ i σ ic δ b = 0 . ( ∗ ) − a n +1 Here σ ab := g ab · det( g ) 1 / ( n +1) should be understood as an element of S 2 M ⊗ (Λ n ) 2 / ( n +1) M . In particular, ∂ 2 ∂ x a σ bc + Γ b ad σ dc + Γ c ∇ a σ bc = da σ bd n + 1Γ d da σ bc − � �� � � �� � Usual covariant derivative addition coming from volume form
Theorem (Eastwood-Matveev 2006; special cases known to Liouville 1889, Sinjukov 1961, Bolsinov-Matveev 2003) The Levi-Civita connection of g lies in a projective class of a connection Γ i jk if and only if σ ab := g ab · det( g ) 1 / ( n +1) is a solution of � ∇ a σ bc � � � 1 ∇ i σ ib δ c a + ∇ i σ ic δ b = 0 . ( ∗ ) − a n +1 Here σ ab := g ab · det( g ) 1 / ( n +1) should be understood as an element of S 2 M ⊗ (Λ n ) 2 / ( n +1) M . In particular, ∂ 2 ∂ x a σ bc + Γ b ad σ dc + Γ c ∇ a σ bc = da σ bd n + 1Γ d da σ bc − � �� � � �� � Usual covariant derivative addition coming from volume form � � n 2 ( n +1) The equations ( ∗ ) is a system of − n linear PDEs of the first 2 order on n ( n +1) unknown components of σ . 2
Theorem (Eastwood-Matveev 2006; special cases known to Liouville 1889, Sinjukov 1961, Bolsinov-Matveev 2003) The Levi-Civita connection of g lies in a projective class of a connection Γ i jk if and only if σ ab := g ab · det( g ) 1 / ( n +1) is a solution of � ∇ a σ bc � � � 1 ∇ i σ ib δ c a + ∇ i σ ic δ b = 0 . ( ∗ ) − a n +1 Here σ ab := g ab · det( g ) 1 / ( n +1) should be understood as an element of S 2 M ⊗ (Λ n ) 2 / ( n +1) M . In particular, ∂ 2 ∂ x a σ bc + Γ b ad σ dc + Γ c ∇ a σ bc = da σ bd n + 1Γ d da σ bc − � �� � � �� � Usual covariant derivative addition coming from volume form � � n 2 ( n +1) The equations ( ∗ ) is a system of − n linear PDEs of the first 2 order on n ( n +1) unknown components of σ . 2
Theorem (Matveev-Rosemann 2011/independently Calderbank The Levi-Civita connection of g on ( M 2 n , J ) lies in a h -projective 2011) jk if and only if σ ab := g ab · det( g ) 1 / (2 n +2) is a class of a connection Γ i solution of k ∇ ℓ σ ℓ c + δ c a ∇ ℓ σ ℓ b + J b m ∇ ℓ σ ℓ m + J c ∇ a σ bc − 1 2 n ( δ b a J c a J b m ∇ ℓ σ ℓ m ) = 0 . ( ∗∗ )
Theorem (Matveev-Rosemann 2011/independently Calderbank The Levi-Civita connection of g on ( M 2 n , J ) lies in a h -projective 2011) jk if and only if σ ab := g ab · det( g ) 1 / (2 n +2) is a class of a connection Γ i solution of k ∇ ℓ σ ℓ c + δ c a ∇ ℓ σ ℓ b + J b m ∇ ℓ σ ℓ m + J c ∇ a σ bc − 1 2 n ( δ b a J c a J b m ∇ ℓ σ ℓ m ) = 0 . ( ∗∗ ) Here σ ab := g ab · det( g ) 1 / (2 n +2) should be understood as an element of S 2 M ⊗ (Λ n ) 1 / ( n +1) M . In particular, ∂ 1 ∂ x a σ bc + Γ b ad σ dc + Γ c ∇ a σ bc = da σ bd n + 1Γ d da σ bc − � �� � � �� � Usual covariant derivative addition coming from volume form
Properties and advantages of the equations ( ∗ ) (resp. ( ∗∗ ))
Properties and advantages of the equations ( ∗ ) (resp. ( ∗∗ )) 1. They are linear PDE systems of finite type (close after two prolongations). In the projective case, there exists at most ( n +1)( n +2) -dimensional space of solutions. In the h -projective case, 2 there exists at most ( n + 1) 2 -dimensional space of solutions.
Properties and advantages of the equations ( ∗ ) (resp. ( ∗∗ )) 1. They are linear PDE systems of finite type (close after two prolongations). In the projective case, there exists at most ( n +1)( n +2) -dimensional space of solutions. In the h -projective case, 2 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 connection withing the projective (resp. h -projective) class.
Properties and advantages of the equations ( ∗ ) (resp. ( ∗∗ )) 1. They are linear PDE systems of finite type (close after two prolongations). In the projective case, there exists at most ( n +1)( n +2) -dimensional space of solutions. In the h -projective case, 2 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 connection withing 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 metric of dimension ≥ 4 is � �� � will be explained projectively rigid (i.e., every metric projectively equivalent to g is proportional to g) .
Theorem (Matveev 2011) Almost every metric of dimension ≥ 4 is � �� � will be explained projectively rigid (i.e., every metric projectively equivalent to g is proportional to g) .
Theorem (Matveev 2011) Almost every metric of dimension ≥ 4 is � �� � will be explained projectively rigid (i.e., every metric projectively equivalent to g is proportional to g) . What we understand under almost every?
Theorem (Matveev 2011) Almost every metric of dimension ≥ 4 is � �� � will be explained projectively rigid (i.e., every metric projectively equivalent to g is proportional to g) . What we understand under almost every? We consider the standard uniform C 2 − topology: the metric g is ε − close to the metric ¯ g in this topology, if the components of g and their first and second derivatives are ε − close to that of ¯ g .
Theorem (Matveev 2011) Almost every metric of dimension ≥ 4 is � �� � will be explained projectively rigid (i.e., every metric projectively equivalent to g is proportional to g) . What we understand under almost every? We consider the standard uniform C 2 − topology: the metric g is ε − close to the metric ¯ g in this topology, if the components of g and their first and second derivatives are ε − close to that of ¯ g . ‘Almost every’ in the statement of Theorem above should be understood as the set of geodesically ri- Arbitrary gid 4D metrics contains an Open subset of projectively metric g rigid metrics open everywhere dense (in C 2 -topology) subset. Arbitrary small neighborhood of g in the space of all metrics on U with C² topology
Theorem (Matveev 2011) Almost every metric of dimension ≥ 4 is � �� � will be explained projectively rigid (i.e., every metric projectively equivalent to g is proportional to g) . What we understand under almost every? We consider the standard uniform C 2 − topology: the metric g is ε − close to the metric ¯ g in this topology, if the components of g and their first and second derivatives are ε − close to that of ¯ g . ‘Almost every’ in the statement of Theorem above should be understood as the set of geodesically ri- Arbitrary gid 4D metrics contains an Open subset of projectively metric g rigid metrics open everywhere dense (in C 2 -topology) subset. Arbitrary small neighborhood of g in the space of all metrics on U with C² topology The result survives in dim 3, if we replace the uniform C 2 − topology by the uniform C 3 -topology (based on Sinjukov 1954). In dim 2, the result is again true, if we replace the uniform C 2 − topology by the uniform C 6 -topology (based on nontrivial calculations of Kruglikov 2009 and Bryant–Dunajski–Eastwood 2011).
Theorem (Matveev 2011) Almost every metric of dimension ≥ 4 is � �� � will be explained projectively rigid (i.e., every metric projectively equivalent to g is proportional to g) . What we understand under almost every? We consider the standard uniform C 2 − topology: the metric g is ε − close to the metric ¯ g in this topology, if the components of g and their first and second derivatives are ε − close to that of ¯ g . ‘Almost every’ in the statement of Theorem above should be understood as the set of geodesically ri- Arbitrary gid 4D metrics contains an Open subset of projectively metric g rigid metrics open everywhere dense (in C 2 -topology) subset. Arbitrary small neighborhood of g in the space of all metrics on U with C² topology The result survives in dim 3, if we replace the uniform C 2 − topology by the uniform C 3 -topology (based on Sinjukov 1954). In dim 2, the result is again true, if we replace the uniform C 2 − topology by the uniform C 6 -topology (based on nontrivial calculations of Kruglikov 2009 and Bryant–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 admitting projective or h -projective symmetry.
Main Theorems Thus, “most” metrics do not admit projective or h -projective symmetry; but still locally there exist tons of examples of metrics admitting projective or h -projective symmetry. Theorem (“Lichnerowicz-Obata conjecture”, Matveev JDG 2007) . Let ( M , g ) be a compact, connected Riemannian manifold of real dimension n ≥ 2. If ( M , g ) cannot be covered by ( S n , c · g round) for some c > 0, then Iso 0 = Pro 0 .
Main Theorems Thus, “most” metrics do not admit projective or h -projective symmetry; but still locally there exist tons of examples of metrics admitting projective or h -projective symmetry. Theorem (“Lichnerowicz-Obata conjecture”, Matveev JDG 2007) . Let ( M , g ) be a compact, connected Riemannian manifold of real dimension n ≥ 2. If ( M , g ) cannot be covered by ( S n , c · g round) for some c > 0, then Iso 0 = Pro 0 .
Main Theorems Thus, “most” metrics do not admit projective or h -projective symmetry; but still locally there exist tons of examples of metrics admitting projective or h -projective symmetry. Theorem (“Lichnerowicz-Obata conjecture”, Matveev JDG 2007) . Let ( M , g ) be a compact, connected Riemannian manifold of real dimension n ≥ 2. If ( M , g ) cannot be covered by ( S n , c · g round) for some c > 0, then Iso 0 = Pro 0 . Theorem (“Yano-Obata conjecture”, Matveev-Rosemann JDG (to appear in 2013), +Fedorova + Kiosak PLM 2012) . Let ( M , g , J ) be a compact, connected Riemannian K¨ ahler manifold of real dimension 2 n ≥ 4. If ( M , g , J ) is not ( C P ( n ) , c · g FS , J standard ) for some c > 0, then Iso 0 = HPro 0 . (Here g FS is the Fubini-Studi metric).
Special cases were proved before by French, Japanese and Soviet geometry schools. Lichnerowciz-Obata conjecture France Japan Soviet Union (Lichnerowicz) (Yano, Obata, Tanno) (Raschewskii) Solodovnikov (1956) Couty (1961) proved Yamauchi (1974) pro- proved the conjecture the conjecture assu- ved the conjecture as- assuming that all ob- ming that g is Einstein suming that the scalar jects are real analytic or K¨ ahler curvature is constant and that n ≥ 3.
Special cases were proved before by French, Japanese and Soviet geometry schools. Lichnerowciz-Obata conjecture France Japan Soviet Union (Lichnerowicz) (Yano, Obata, Tanno) (Raschewskii) Solodovnikov (1956) Couty (1961) proved Yamauchi (1974) pro- proved the conjecture the conjecture assu- ved the conjecture as- assuming that all ob- ming that g is Einstein suming that the scalar jects are real analytic or K¨ ahler curvature is constant and 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 S n and CP ( n ) the groups of projective resp. h -projective transformations are much bigger than the groups of isometries.
In S n and CP ( n ) the groups of projective resp. h -projective transformations are much bigger than the groups of isometries. We consider the standard S n ⊂ R n +1 with the induced metric.
In S n and CP ( n ) the groups of projective resp. h -projective transformations are much bigger than the groups of isometries. We consider the standard S n ⊂ R n +1 with the induced metric. Fact. Geodesics of the sphere are the great circles, that are the intersec- tions of the 2-planes containing the center of the sphere with the sphere.
In S n and CP ( n ) the groups of projective resp. h -projective transformations are much bigger than the groups of isometries. We consider the standard S n ⊂ R n +1 with the induced metric. Fact. Geodesics of the sphere are the great circles, that are the intersec- tions of the 2-planes containing the center of the sphere with the sphere. Proof. We consider the reflection with respect to the corresponding 2-plane. It is an isometry of the sphere; its sets of fixed points is the great circle and is totally geodesics.
In S n and CP ( n ) the groups of projective resp. h -projective transformations are much bigger than the groups of isometries. We consider the standard S n ⊂ R n +1 with the induced metric. Fact. Geodesics of the sphere are the great circles, that are the intersec- tions of the 2-planes containing the center of the sphere with the sphere. Proof. We consider the reflection with respect to the corresponding 2-plane. It is an isometry of the sphere; its sets of fixed points is the great circle and is totally geodesics. Indeed, If this is geodesic would a geodesic tangent to the great circle leave it, it would give a contra- diction with the uniqueness theorem for solutions of ODE d 2 x a dx b dx c dt = α ( t ) dx a dt 2 + Γ a then ist reflection bc dt dt Greate circle is also a geodesic (with any fixed α ). contradicting the uniqence
Beltrami example
Beltrami example Beltrami (1865) observed:
Beltrami example Beltrami (1865) observed: we construct → a : S n → S n , a ( x ) := A ( x ) For every A ∈ SL ( n + 1) − − − − − − − − | A ( x ) |
Beltrami example Beltrami (1865) observed: we construct → a : S n → S n , a ( x ) := A ( x ) For every A ∈ SL ( n + 1) − − − − − − − − | A ( x ) | ◮ a is a diffeomorphism
Beltrami example Beltrami (1865) observed: we construct → a : S n → S n , a ( x ) := A ( x ) For every A ∈ SL ( n + 1) − − − − − − − − | A ( x ) | ◮ a is a diffeomorphism ◮ a takes great circles (geodesics) to great circles (geodesics)
Beltrami example Beltrami (1865) observed: we construct → a : S n → S n , a ( x ) := A ( x ) For every A ∈ SL ( n + 1) − − − − − − − − | A ( x ) | ◮ a is a diffeomorphism ◮ a takes great circles (geodesics) to great circles (geodesics) ◮ a is an isometry iff A ∈ O ( n + 1).
Beltrami example Beltrami (1865) observed: we construct → a : S n → S n , a ( x ) := A ( x ) For every A ∈ SL ( n + 1) − − − − − − − − | A ( x ) | ◮ a is a diffeomorphism ◮ a takes great circles (geodesics) to great circles (geodesics) ◮ a is an isometry iff A ∈ O ( n + 1). Thus, Sl ( n + 1) acts by projective transformations on S n . We see that Proj 0 is bigger than Iso 0 = SO ( n + 1)
In CP ( n ), the situation is essentially the same Fact. A curve on ( CP ( n ) , g FS , J ) is h − planar, if and only if it lies on a projective line (which are totally geodesic complex surfaces in CP ( n ) homeomorphic to the sphere).
In CP ( n ), the situation is essentially the same Fact. A curve on ( CP ( n ) , g FS , J ) is h − planar, if and only if it lies on a projective line (which are totally geodesic complex surfaces in CP ( 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, every h -planar curve whose tangent vector is tangent to a projective line stays on the projective line by the uniqueness of the solutions of a system of ODE d 2 x a dx b dx c dt = α ( t ) dx a dt + β ( t ) dx k dt 2 + Γ a dt J a k (for fixed bc dt α, β ).
In CP ( n ), the situation is essentially the same Fact. A curve on ( CP ( n ) , g FS , J ) is h − planar, if and only if it lies on a projective line (which are totally geodesic complex surfaces in CP ( 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, every h -planar curve whose tangent vector is tangent to a projective line stays on the projective line by the uniqueness of the solutions of a system of ODE d 2 x a dx b dx c dt = α ( t ) dx a dt + β ( t ) dx k dt 2 + Γ a dt J a k (for fixed bc dt α, β ). From the other side, since the tangent space TL ⊂ TCP ( n ) of every projective line is J − invariant, every curve lying on the projective line is h -planar ( because ∇ ˙ γ ˙ γ ∈ TL and is therefore a linear combination of ˙ γ and J (˙ γ ) since TL is two-dimensional and J -invariant).
In CP ( n ), the situation is essentially the same Fact. A curve on ( CP ( n ) , g FS , J ) is h − planar, if and only if it lies on a projective line (which are totally geodesic complex surfaces in CP ( 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, every h -planar curve whose tangent vector is tangent to a projective line stays on the projective line by the uniqueness of the solutions of a system of ODE d 2 x a dx b dx c dt = α ( t ) dx a dt + β ( t ) dx k dt 2 + Γ a dt J a k (for fixed bc dt α, β ). From the other side, since the tangent space TL ⊂ TCP ( n ) of every projective line is J − invariant, every curve lying on the projective line is h -planar ( because ∇ ˙ γ ˙ γ ∈ TL and is therefore a linear combination of ˙ γ and J (˙ γ ) since TL is two-dimensional and J -invariant). Corollary ( h -projective analog of Beltrami Example). The group of h -projective transformations is SL ( n + 1 , C ) and is much bigger than the group of isometries which is SU ( n + 1).
Question. The main results and, actually, most questions asked by classics, do not really require projective and h -projective structures (since all the questions are about metrics). Why we introduced them?
Question. The main results and, actually, most questions asked by classics, do not really require projective and h -projective structures (since all the questions are about metrics). Why we introduced 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 vector field unless g has constant sectional curvature (resp. constant holomorphic sectional curvature).
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 vector field unless g has constant sectional curvature (resp. constant holomorphic sectional curvature). 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. 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 vector field unless g has constant sectional curvature (resp. constant holomorphic sectional curvature). 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 , every two projective ( h -projectively, resp.) metrics are proportional. Then, a projective ( h − projective) vector field is a infinitesimal homothety. Since our manifold is closed, every homothety is isometry so our vector field is a Killing.
If the degree of mobility of the projective structure is 1 , every two projective ( h -projectively, resp.) metrics are proportional. Then, a projective ( h − projective) vector field is a infinitesimal homothety. Since our manifold is closed, every homothety is isometry 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-Matveev 2010/Matveev-Mounoud 2011 for projective structures; Fedorova-Kiosak-Matveev-Rosemann for h -projective structures). If the degree of mobility ≥ 3, the LO and YO conjectures hold (even in the pseudo-Riemannian case).
If the degree of mobility of the projective structure is 1 , every two projective ( h -projectively, resp.) metrics are proportional. Then, a projective ( h − projective) vector field is a infinitesimal homothety. Since our manifold is closed, every homothety is isometry 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-Matveev 2010/Matveev-Mounoud 2011 for projective structures; Fedorova-Kiosak-Matveev-Rosemann for h -projective structures). 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 the methods of the next part of my talk. I will touch them if I have time
If the degree of mobility of the projective structure is 1 , every two projective ( h -projectively, resp.) metrics are proportional. Then, a projective ( h − projective) vector field is a infinitesimal homothety. Since our manifold is closed, every homothety is isometry 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-Matveev 2010/Matveev-Mounoud 2011 for projective structures; Fedorova-Kiosak-Matveev-Rosemann for h -projective structures). 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 the methods of the next part of my talk. I will touch them if I have time Thus, the only remaining case in when the degree of mobility is 2
The case degree of mobility =2 Let Sol be the space of solutions of the equation ( ∗ ) or ( ∗∗ ); it is a two-dimensional vector space. Let v is a projective (resp. h -projective) vector field.
The case degree of mobility =2 Let Sol be the space of solutions of the equation ( ∗ ) or ( ∗∗ ); it is a two-dimensional vector space. Let v is a projective (resp. h -projective) vector field. Important observation. L v : Sol → Sol , where L v is the Lie derivative.
The case degree of mobility =2 Let Sol be the space of solutions of the equation ( ∗ ) or ( ∗∗ ); it is a two-dimensional vector space. Let v is a projective (resp. h -projective) vector field. Important observation. L v : Sol → Sol , where L v is the Lie derivative. Proof. The equations ( ∗ ) (resp. ( ∗∗ )) are projective (resp. h -projective) invariant.
The case degree of mobility =2 Let Sol be the space of solutions of the equation ( ∗ ) or ( ∗∗ ); it is a two-dimensional vector space. Let v is a projective (resp. h -projective) vector field. Important observation. L v : Sol → Sol , where L v is the Lie derivative. Proof. The equations ( ∗ ) (resp. ( ∗∗ )) are projective (resp. h -projective) ∂ invariant. Then, in a coordinate system such that v = ∂ x 1 the coefficients in the equations do not depend on the x 1 -coordinate.
The case degree of mobility =2 Let Sol be the space of solutions of the equation ( ∗ ) or ( ∗∗ ); it is a two-dimensional vector space. Let v is a projective (resp. h -projective) vector field. Important observation. L v : Sol → Sol , where L v is the Lie derivative. Proof. The equations ( ∗ ) (resp. ( ∗∗ )) are projective (resp. h -projective) ∂ invariant. Then, in a coordinate system such that v = ∂ x 1 the coefficients in the equations do not depend on the x 1 -coordinate. Then, for every solution σ ij its x 1 -derivative ∂ ∂ x 1 σ ij , which is precisely the Lie derivative, is also a solution, .
The case degree of mobility =2 Let Sol be the space of solutions of the equation ( ∗ ) or ( ∗∗ ); it is a two-dimensional vector space. Let v is a projective (resp. h -projective) vector field. Important observation. L v : Sol → Sol , where L v is the Lie derivative. Proof. The equations ( ∗ ) (resp. ( ∗∗ )) are projective (resp. h -projective) ∂ invariant. Then, in a coordinate system such that v = ∂ x 1 the coefficients in the equations do not depend on the x 1 -coordinate. Then, for every solution σ ij its x 1 -derivative ∂ ∂ x 1 σ ij , 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 ): � � � � � � L v σ = λσ L v σ = λσ + µ ¯ σ L v σ = λσ +¯ σ L v ¯ σ = µ ¯ σ L v ¯ σ = − µσ + λ ¯ σ L v ¯ σ = λ ¯ σ
We obtained that the derivatives of σ, ¯ σ along the flow of v are given by � � � � � � L v σ = λσ L v σ = λσ + µ ¯ σ L v σ = λσ +¯ σ . L v ¯ σ = µ ¯ σ L v ¯ σ = − µσ + λ ¯ σ L v ¯ σ = λ ¯ σ
We obtained that the derivatives of σ, ¯ σ along the flow of v are given by � � � � � � L v σ = λσ L v σ = λσ + µ ¯ σ L v σ = λσ +¯ σ . L v ¯ σ = µ ¯ σ L v ¯ σ = − µσ + λ ¯ σ L v ¯ σ = λ ¯ σ Thus, the evolution of the solutions along the flow φ t of v are given by the matrices � � φ ∗ = e λ t σ t σ e µ t ¯ φ ∗ t ¯ σ = σ � φ ∗ = e λ t cos( µ t ) σ + e λ t sin( µ t )¯ � t σ σ = − e λ t sin( µ t ) σ + e λ t cos( µ t )¯ φ ∗ t ¯ σ σ � + te λ t ¯ � φ ∗ = e λ t σ t σ σ . e λ t ¯ φ ∗ t ¯ σ = σ We will consider all these three cases separately.
The simplest case is when the evolution is given by � φ ∗ = e λ t cos( µ t ) σ + e λ t sin( µ t )¯ � t σ σ . = − e λ t sin( µ t ) σ + e λ t cos( µ t )¯ φ ∗ t ¯ σ σ
The simplest case is when the evolution is given by � φ ∗ = e λ t cos( µ t ) σ + e λ t sin( µ t )¯ � t σ σ . = − e λ t sin( µ t ) σ + e λ t cos( µ t )¯ φ ∗ t ¯ σ σ Suppose our metrics correspond to the element a σ + b ¯ σ .
The simplest case is when the evolution is given by � φ ∗ = e λ t cos( µ t ) σ + e λ t sin( µ t )¯ � t σ σ . = − e λ t sin( µ t ) σ + e λ t cos( µ t )¯ φ ∗ t ¯ σ σ Suppose our metrics correspond to the element a σ + b ¯ σ . Its evolution is given by = a ( e λ t cos( µ t ) σ + e λ t sin( µ t )¯ φ ∗ t ( a σ + b ¯ σ ) σ ) + b ( − e λ t sin( µ t ) σ + e λ t cos( µ t )¯ σ ) = e λ t √ a 2 + b 2 (cos( µ t + α ) σ + sin( µ t + α )¯ σ ) , √ a 2 + b 2 )). where α = arccos( a / (
The simplest case is when the evolution is given by � φ ∗ = e λ t cos( µ t ) σ + e λ t sin( µ t )¯ � t σ σ . = − e λ t sin( µ t ) σ + e λ t cos( µ t )¯ φ ∗ t ¯ σ σ Suppose our metrics correspond to the element a σ + b ¯ σ . Its evolution is given by = a ( e λ t cos( µ t ) σ + e λ t sin( µ t )¯ φ ∗ t ( a σ + b ¯ σ ) σ ) + b ( − e λ t sin( µ t ) σ + e λ t cos( µ t )¯ σ ) = e λ t √ a 2 + b 2 (cos( µ t + α ) σ + sin( µ t + α )¯ σ ) , √ a 2 + b 2 )). where α = arccos( a / ( Now, we use that the metric is Riemannian. Then, for any point x there exists a basis in T x M such that σ and ¯ σ are given by diagonal matrices: σ = diag ( s 1 , s 2 , ... ) and ¯ σ = diag (¯ s 1 , ¯ s 2 , ... ).
The simplest case is when the evolution is given by � φ ∗ = e λ t cos( µ t ) σ + e λ t sin( µ t )¯ � t σ σ . = − e λ t sin( µ t ) σ + e λ t cos( µ t )¯ φ ∗ t ¯ σ σ Suppose our metrics correspond to the element a σ + b ¯ σ . Its evolution is given by = a ( e λ t cos( µ t ) σ + e λ t sin( µ t )¯ φ ∗ t ( a σ + b ¯ σ ) σ ) + b ( − e λ t sin( µ t ) σ + e λ t cos( µ t )¯ σ ) = e λ t √ a 2 + b 2 (cos( µ t + α ) σ + sin( µ t + α )¯ σ ) , √ a 2 + b 2 )). where α = arccos( a / ( Now, we use that the metric is Riemannian. Then, for any point x there exists a basis in T x M such that σ and ¯ σ are given by diagonal matrices: σ = diag ( s 1 , s 2 , ... ) and ¯ σ = diag (¯ s 1 , ¯ s 2 , ... ). Then, φ ∗ t ( a σ + b ¯ σ ) at this point is also diagonal with the i th element e λ t √ a 2 + b 2 (cos( µ t + α ) s i + sin( µ t + α )¯ s i ).
The simplest case is when the evolution is given by � φ ∗ = e λ t cos( µ t ) σ + e λ t sin( µ t )¯ � t σ σ . = − e λ t sin( µ t ) σ + e λ t cos( µ t )¯ φ ∗ t ¯ σ σ Suppose our metrics correspond to the element a σ + b ¯ σ . Its evolution is given by = a ( e λ t cos( µ t ) σ + e λ t sin( µ t )¯ φ ∗ t ( a σ + b ¯ σ ) σ ) + b ( − e λ t sin( µ t ) σ + e λ t cos( µ t )¯ σ ) = e λ t √ a 2 + b 2 (cos( µ t + α ) σ + sin( µ t + α )¯ σ ) , √ a 2 + b 2 )). where α = arccos( a / ( Now, we use that the metric is Riemannian. Then, for any point x there exists a basis in T x M such that σ and ¯ σ are given by diagonal matrices: σ = diag ( s 1 , s 2 , ... ) and ¯ σ = diag (¯ s 1 , ¯ s 2 , ... ). Then, φ ∗ t ( a σ + b ¯ σ ) at this point is also diagonal with the i th element e λ t √ a 2 + b 2 (cos( µ t + α ) s i + sin( µ t + α )¯ s i ). Clearly, for a certain t we have that φ ∗ t ( a σ + b ¯ σ ) is degenerate which contradicts the assumption,
The proof is is similar when the evolution is given by � + te λ t ¯ � φ ∗ = e λ t σ t σ σ . e λ t ¯ φ ∗ t ¯ σ = σ
The proof is is similar when the evolution is given by � + te λ t ¯ � φ ∗ = e λ t σ t σ σ . e λ t ¯ φ ∗ t ¯ σ = σ We again suppose that our metrics correspond to the element a σ + b ¯ σ .
The proof is is similar when the evolution is given by � + te λ t ¯ � φ ∗ = e λ t σ t σ σ . e λ t ¯ φ ∗ t ¯ σ = σ We again suppose that our metrics correspond to the element a σ + b ¯ σ . Its evolution is given by σ ) + b ( e λ t ¯ φ ∗ = a ( e λ t σ + e λ t t ¯ t ( a σ + b ¯ σ ) σ ) = e λ t ( a σ + ( b + at )¯ σ ) .
The proof is is similar when the evolution is given by � + te λ t ¯ � φ ∗ = e λ t σ t σ σ . e λ t ¯ φ ∗ t ¯ σ = σ We again suppose that our metrics correspond to the element a σ + b ¯ σ . Its evolution is given by σ ) + b ( e λ t ¯ φ ∗ = a ( e λ t σ + e λ t t ¯ t ( a σ + b ¯ σ ) σ ) = e λ t ( a σ + ( b + at )¯ σ ) . We again see that unless a � = 0 there exists t such that φ ∗ t ( a σ + b ¯ σ ) is degenerate which contradicts the assumption.
The proof is is similar when the evolution is given by � + te λ t ¯ � φ ∗ = e λ t σ t σ σ . e λ t ¯ φ ∗ t ¯ σ = σ We again suppose that our metrics correspond to the element a σ + b ¯ σ . Its evolution is given by σ ) + b ( e λ t ¯ φ ∗ = a ( e λ t σ + e λ t t ¯ t ( a σ + b ¯ σ ) σ ) = e λ t ( a σ + ( b + at )¯ σ ) . We again see that unless a � = 0 there exists t such that φ ∗ t ( a σ + b ¯ σ ) is degenerate which contradicts the assumption. Now, if a = 0, then g corresponds to ¯ σ and v is its Killing vector field,
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