Flag Curvature. Miguel Angel Javaloyes. Cedeira 31 octubre a 1 noviembre. Universidad de Granada. M. A. Javaloyes (*) Flag Curvature 1 / 23
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1 Flag Curvature Miguel Angel Javaloyes Universidad de Granada Cedeira 31 octubre a 1 noviembre M. A. Javaloyes (*) Flag Curvature 1 / 23
2 Chern Connection S.S. Chern ( ) M. A. Javaloyes (*) Flag Curvature 2 / 23
3 Chern Connection π : TM \ {0} M is the natural projection S.S. Chern ( ) M. A. Javaloyes (*) Flag Curvature 2 / 23
4 Chern Connection π : TM \ {0} M is the natural projection now we take the pullback of TM by dπ = π, that is, π TM S.S. Chern ( ) M. A. Javaloyes (*) Flag Curvature 2 / 23
5 Chern Connection π : TM \ {0} M is the natural projection now we take the pullback of TM by dπ = π, that is, π TM We have a metric over this vector bundle given by g ij (x, y)dx i dx j, where g ij (x, y) = 1 2 (F 2 ) 2 y i (x, y) y j S.S. Chern ( ) M. A. Javaloyes (*) Flag Curvature 2 / 23
6 Chern Connection M. A. Javaloyes (*) Flag Curvature 3 / 23
7 Chern Connection The Chern connection is the unique linear connection on π TM whose connection 1-forms ω i j satisfy: M. A. Javaloyes (*) Flag Curvature 3 / 23
8 Chern Connection The Chern connection is the unique linear connection on π TM whose connection 1-forms ω i j satisfy: dx j ω i j = 0 torsion free (1) dg ij g kj ω k i g ik ω k i = 2 F A ijsδy s almost g-compatibility (2) M. A. Javaloyes (*) Flag Curvature 3 / 23
9 Chern Connection The Chern connection is the unique linear connection on π TM whose connection 1-forms ω i j satisfy: dx j ω i j = 0 torsion free (1) dg ij g kj ω k i g ik ω k i = 2 F A ijsδy s almost g-compatibility (2) where δy s are the 1-forms on π TM given as δy s := dy s + N s j dx j, and N i j (x, y) := γ i jk y k 1 F Ai jk γk rsy r y s are the coefficients of the so called nonlinear connection on TM \ 0, and M. A. Javaloyes (*) Flag Curvature 3 / 23
10 Chern Connection The Chern connection is the unique linear connection on π TM whose connection 1-forms ω i j satisfy: dx j ω i j = 0 torsion free (1) dg ij g kj ω k i g ik ω k i = 2 F A ijsδy s almost g-compatibility (2) where δy s are the 1-forms on π TM given as δy s := dy s + N s j dx j, and N i j (x, y) := γ i jk y k 1 F Ai jk γk rsy r y s are the coefficients of the so called nonlinear connection on TM \ 0, and γ i jk(x, y) = 1 ( 2 g is gsj x k g jk x s + g ) ks x j, A ijk (x, y) = F g ij 2 y k = F 3 (F 2 ) 4 y i y j y k, M. A. Javaloyes (*) Flag Curvature 3 / 23
11 Covariant derivatives M. A. Javaloyes (*) Flag Curvature 4 / 23
12 Covariant derivatives The components of the Chern connection are given by: Γ i jk (x, y) = γi jk g il F ( Aljs Nk s A jksni s + A kls Nj s ). that is, ω i j = Γ i jk dx k. M. A. Javaloyes (*) Flag Curvature 4 / 23
13 Covariant derivatives The components of the Chern connection are given by: Γ i jk (x, y) = γi jk g il F ( Aljs Nk s A jksni s + A kls Nj s ). that is, ωj i = Γ i jk dx k. The Chern connection gives two different covariant derivatives: ( ) dw i D T W = + W j T k Γ i jk dt (γ, T ) x i with ref. vector T, γ(t) ( ) dw i D T W = + W j T k Γ i jk dt (γ, W ) x i with ref. vector W. γ(t) M. A. Javaloyes (*) Flag Curvature 4 / 23
14 Other connections M. A. Javaloyes (*) Flag Curvature 5 / 23
15 Other connections Cartan connection: metric compatible but has torsion E. Cartan ( ) M. A. Javaloyes (*) Flag Curvature 5 / 23
16 Other connections Cartan connection: metric compatible but has torsion Hashiguchi connection Masao Hashiguchi E. Cartan ( ) M. A. Javaloyes (*) Flag Curvature 5 / 23
17 Other connections Cartan connection: metric compatible but has torsion Hashiguchi connection Berwald connection: no torsion. Specially good to treat with Finsler spaces of constant flag curvature Masao Hashiguchi Ludwig Berwald 1883 (Prague)-1942 E. Cartan ( ) M. A. Javaloyes (*) Flag Curvature 5 / 23
18 Other connections Cartan connection: metric compatible but has torsion Hashiguchi connection Berwald connection: no torsion. Specially good to treat with Finsler spaces of constant flag curvature Rund connection: coincides with Chern connection Masao Hashiguchi Ludwig Berwald 1883 (Prague)-1942 E. Cartan ( ) Hanno Rund , South Africa M. A. Javaloyes (*) Flag Curvature 5 / 23
19 Curvature 2-forms of the Chern connection The curvature 2-forms of the Chern connection are: Ω i j := dω i j ω k j ω i k M. A. Javaloyes (*) Flag Curvature 6 / 23
20 Curvature 2-forms of the Chern connection The curvature 2-forms of the Chern connection are: Ω i j := dω i j ω k j ω i k It can be expanded as Ω i j := 1 2 R i j kl dx k dx l + Pj i kl dx k δy l F Q j i δy k kl F δy l F M. A. Javaloyes (*) Flag Curvature 6 / 23
21 Curvature 2-forms of the Chern connection The curvature 2-forms of the Chern connection are: Ω i j := dω i j ω k j ω i k It can be expanded as Ω i j := 1 2 R i j kl dx k dx l + Pj i kl dx k δy l F Q j i δy k kl F δy l F From free torsion of the Chern connection Q i j kl = 0 M. A. Javaloyes (*) Flag Curvature 6 / 23
22 Curvature 2-forms of the Chern connection The curvature 2-forms of the Chern connection are: Ω i j := dω i j ω k j ω i k It can be expanded as Ω i j := 1 2 R i j kl dx k dx l + Pj i kl dx k δy l F From free torsion of the Chern connection Q i j kl = 0 M. A. Javaloyes (*) Flag Curvature 6 / 23
23 Curvature 2-forms of the Chern connection The curvature 2-forms of the Chern connection are: Ω i j := dω i j ω k j ω i k It can be expanded as Ω i j := 1 2 R i j kl dx k dx l + Pj i kl dx k δy l F From free torsion of the Chern connection Q i j kl = 0 Rj i kl = δγi jl δx k δγi jk δx k + Γ i hk Γh jl Γi hl Γh jk ( δ δx k = x k N i y i ) M. A. Javaloyes (*) Flag Curvature 6 / 23
24 Curvature 2-forms of the Chern connection The curvature 2-forms of the Chern connection are: Ω i j := dω i j ω k j ω i k It can be expanded as Ω i j := 1 2 R i j kl dx k dx l + Pj i kl dx k δy l F From free torsion of the Chern connection Q i j kl = 0 Rj i kl = δγi jl δx k δγi jk δx k Pj i kl = F Γi jk y l + Γ i hk Γh jl Γi hl Γh jk ( δ δx k = x k N i y i ) M. A. Javaloyes (*) Flag Curvature 6 / 23
25 Bianchi Identities First Bianchi Identity for R Luigi Bianchi ( ) M. A. Javaloyes (*) Flag Curvature 7 / 23
26 Bianchi Identities First Bianchi Identity for R R i j kl + R k i lj + R i l jk = 0 Luigi Bianchi ( ) M. A. Javaloyes (*) Flag Curvature 7 / 23
27 Bianchi Identities First Bianchi Identity for R Rj i kl + R k i lj + R l i jk = 0 Other identities: P i k jl = Pj i kl Luigi Bianchi ( ) M. A. Javaloyes (*) Flag Curvature 7 / 23
28 Bianchi Identities First Bianchi Identity for R Rj i kl + R k i lj + R l i jk = 0 Other identities: Pk i jl = Pj i kl R ijkl + R jikl = 2B ijkl, where B ijkl := A iju R u kl, Ru kl = y j R ijkl = g jµ R µ i kl F R j u kl and Luigi Bianchi ( ) M. A. Javaloyes (*) Flag Curvature 7 / 23
29 Bianchi Identities First Bianchi Identity for R Rj i kl + R k i lj + R l i jk = 0 Other identities: Pk i jl = Pj i kl R ijkl + R jikl = 2B ijkl, where B ijkl := A iju R u kl, Ru kl = y j F R j u kl and R ijkl = g jµ R µ i kl R klji R jikl = (B klji B jikl ) + (B kilj + B ljki ) + (B ilji + B jkil ) Luigi Bianchi ( ) M. A. Javaloyes (*) Flag Curvature 7 / 23
30 Bianchi Identities First Bianchi Identity for R Rj i kl + R k i lj + R l i jk = 0 Other identities: Pk i jl = Pj i kl R ijkl + R jikl = 2B ijkl, where B ijkl := A iju R u kl, Ru kl = y j F R j u kl and R ijkl = g jµ R µ i kl R klji R jikl = (B klji B jikl ) + (B kilj + B ljki ) + (B ilji + B jkil ) Second Bianchi identities: very complicated, mix terms in Rj i kl and Pj i kl Luigi Bianchi ( ) M. A. Javaloyes (*) Flag Curvature 7 / 23
31 Flag Curvature We must fix a flagpole y and then a transverse edge V M. A. Javaloyes (*) Flag Curvature 8 / 23
32 Flag Curvature We must fix a flagpole y and then a transverse edge V K(y, V ) := V i (y j R jikl y l )V k g(y, y)g(v, V ) g(y, V ) 2 M. A. Javaloyes (*) Flag Curvature 8 / 23
33 Flag Curvature We must fix a flagpole y and then a transverse edge V K(y, V ) := V i (y j R jikl y l )V k g(y, y)g(v, V ) g(y, V ) 2 We can change V by W = αv + βy, that is, K(y, W ) = K(y, V ). M. A. Javaloyes (*) Flag Curvature 8 / 23
34 Flag Curvature We must fix a flagpole y and then a transverse edge V K(y, V ) := V i (y j R jikl y l )V k g(y, y)g(v, V ) g(y, V ) 2 We can change V by W = αv + βy, that is, K(y, W ) = K(y, V ). We obtain the same quantity with the other connections (Cartan, Berwald, Hasiguchi...) M. A. Javaloyes (*) Flag Curvature 8 / 23
35 Computing Flag curvature M. A. Javaloyes (*) Flag Curvature 9 / 23
36 Computing Flag curvature G i := γ i jk y j y k (spray coefficients) M. A. Javaloyes (*) Flag Curvature 9 / 23
37 Computing Flag curvature G i := γ i jk y j y k (spray coefficients) 2F 2 R i k = 2(G i ) x k 1 2 (G i ) y j (G j ) y k y j (G i ) y k x j + G j (G i ) y k y j M. A. Javaloyes (*) Flag Curvature 9 / 23
38 Computing Flag curvature G i := γ i jk y j y k (spray coefficients) 2F 2 R i k = 2(G i ) x k 1 2 (G i ) y j (G j ) y k y j (G i ) y k x j + G j (G i ) y k y j K(y, V ) = K(l, V ) = V i (R i k )V k g(v,v ) g(l,v ) 2, where l = y/f. f we consider F (x, y) = y, y + df [y], with, the Euclidean metric, hen M. A. Javaloyes (*) Flag Curvature 9 / 23
39 Computing Flag curvature G i := γ i jk y j y k (spray coefficients) 2F 2 R i k = 2(G i ) x k 1 2 (G i ) y j (G j ) y k y j (G i ) y k x j + G j (G i ) y k y j K(y, V ) = K(l, V ) = V i (R i k )V k g(v,v ) g(l,v ) 2, where l = y/f. f we consider F (x, y) = y, y + df [y], with, the Euclidean metric, hen G i = 1 F f x j x k y j y k, very simple!!! M. A. Javaloyes (*) Flag Curvature 9 / 23
40 Computing Flag curvature G i := γ i jk y j y k (spray coefficients) 2F 2 R i k = 2(G i ) x k 1 2 (G i ) y j (G j ) y k y j (G i ) y k x j + G j (G i ) y k y j K(y, V ) = K(l, V ) = V i (R i k )V k g(v,v ) g(l,v ) 2, where l = y/f. f we consider F (x, y) = y, y + df [y], with, the Euclidean metric, hen G i = 1 F f x j x k y j y k, very simple!!! K(y, V ) = K(x, y) = 3 4F 4 (f x i x j y i y j ) 2 1 2F 3 (f x i x j x k y i y j y k ) M. A. Javaloyes (*) Flag Curvature 9 / 23
41 Computing Flag curvature G i := γ i jk y j y k (spray coefficients) 2F 2 R i k = 2(G i ) x k 1 2 (G i ) y j (G j ) y k y j (G i ) y k x j + G j (G i ) y k y j K(y, V ) = K(l, V ) = V i (R i k )V k g(v,v ) g(l,v ) 2, where l = y/f. f we consider F (x, y) = y, y + df [y], with, the Euclidean metric, hen G i = 1 F f x j x k y j y k, very simple!!! K(y, V ) = K(x, y) = 3 4F 4 (f x i x j y i y j ) 2 1 2F 3 (f x i x j x k y i y j y k ) the flag curvature does not depend on the transverse edge!! it is scalar M. A. Javaloyes (*) Flag Curvature 9 / 23
42 Finsler metric with constant flag curvature M. A. Javaloyes (*) Flag Curvature 10 / 23
43 Finsler metric with constant flag curvature The complete classification is an open problem, no Hopf s theorem!!! M. A. Javaloyes (*) Flag Curvature 10 / 23
44 Finsler metric with constant flag curvature The complete classification is an open problem, no Hopf s theorem!!! In the class of Randers metrics there does exist a classification after a long story M. A. Javaloyes (*) Flag Curvature 10 / 23
45 Finsler metric with constant flag curvature The complete classification is an open problem, no Hopf s theorem!!! In the class of Randers metrics there does exist a classification after a long story In 1977 Yasuda and Shimada publishes a paper with a characterization of Randers metrics of scalar flag curvature Hiroshi Yasuda ( ) M. A. Javaloyes (*) Flag Curvature 10 / 23
46 Finsler metric with constant flag curvature The complete classification is an open problem, no Hopf s theorem!!! In the class of Randers metrics there does exist a classification after a long story In 1977 Yasuda and Shimada publishes a paper with a characterization of Randers metrics of scalar flag curvature As a particular case they obtain the Randers metrics of constant flag curvature Hiroshi Yasuda ( ) M. A. Javaloyes (*) Flag Curvature 10 / 23
47 Finsler metric with constant flag curvature The complete classification is an open problem, no Hopf s theorem!!! In the class of Randers metrics there does exist a classification after a long story In 1977 Yasuda and Shimada publishes a paper with a characterization of Randers metrics of scalar flag curvature As a particular case they obtain the Randers metrics of constant flag curvature Shibata-Kitayama in 1984 and Matsumoto in 1989 obtain alternate derivations of the Yasuda-Shimada theorem Hiroshi Yasuda ( ) Makoto Matsumoto M. A. Javaloyes (*) Flag Curvature 10 / 23
48 Finsler metric with constant flag curvature The complete classification is an open problem, no Hopf s theorem!!! In the class of Randers metrics there does exist a classification after a long story In 1977 Yasuda and Shimada publishes a paper with a characterization of Randers metrics of scalar flag curvature As a particular case they obtain the Randers metrics of constant flag curvature Shibata-Kitayama in 1984 and Matsumoto in 1989 obtain alternate derivations of the Yasuda-Shimada theorem In summer 2000, P. Antonelli asks if Yasuda-Shimada theorem is indeed correct Hiroshi Yasuda ( ) Makoto Matsumoto M. A. Javaloyes (*) Flag Curvature 10 / 23
49 Constant flag curvature and Zermelo metrics M. A. Javaloyes (*) Flag Curvature 11 / 23
50 Constant flag curvature and Zermelo metrics In the academic year of Colleen Robles (a graduate student) and David Bao begins to work in a geometrical proof of Yasuda-Shimada theorem M. A. Javaloyes (*) Flag Curvature 11 / 23
51 Constant flag curvature and Zermelo metrics In the academic year of Colleen Robles (a graduate student) and David Bao begins to work in a geometrical proof of Yasuda-Shimada theorem In 17th may 2001 Z. Shen phones D. Bao describing a counterexample to Yasuda-Shimada he found when working with Zermelo metrics M. A. Javaloyes (*) Flag Curvature 11 / 23
52 Constant flag curvature and Zermelo metrics In the academic year of Colleen Robles (a graduate student) and David Bao begins to work in a geometrical proof of Yasuda-Shimada theorem In 17th may 2001 Z. Shen phones D. Bao describing a counterexample to Yasuda-Shimada he found when working with Zermelo metrics In the same year D. Bao-C. Robles and Matsumoto find independently the correct version of Yasuda-Shimada theorem. M. A. Javaloyes (*) Flag Curvature 11 / 23
53 Constant flag curvature and Zermelo metrics In the academic year of Colleen Robles (a graduate student) and David Bao begins to work in a geometrical proof of Yasuda-Shimada theorem In 17th may 2001 Z. Shen phones D. Bao describing a counterexample to Yasuda-Shimada he found when working with Zermelo metrics In the same year D. Bao-C. Robles and Matsumoto find independently the correct version of Yasuda-Shimada theorem. Still no classification (solutions h + h(w, v) must have a h-riemannian curvature related with the module of a h-killing field W ) M. A. Javaloyes (*) Flag Curvature 11 / 23
54 Constant flag curvature and Zermelo metrics In the academic year of Colleen Robles (a graduate student) and David Bao begins to work in a geometrical proof of Yasuda-Shimada theorem In 17th may 2001 Z. Shen phones D. Bao describing a counterexample to Yasuda-Shimada he found when working with Zermelo metrics In the same year D. Bao-C. Robles and Matsumoto find independently the correct version of Yasuda-Shimada theorem. Still no classification (solutions h + h(w, v) must have a h-riemannian curvature related with the module of a h-killing field W ) Finally they perceive that when considering Zermelo expression of Randers metrics the geometry comes out M. A. Javaloyes (*) Flag Curvature 11 / 23
55 Flag constant curvature and stationary spacetimes M. A. Javaloyes (*) Flag Curvature 12 / 23
56 Flag constant curvature and stationary spacetimes Zermelo metric: 1 α g(v, v) + 1 α 2 g(w, v)2 1 g(w, v), α where α = 1 g(w, W ). M. A. Javaloyes (*) Flag Curvature 12 / 23
57 Flag constant curvature and stationary spacetimes Zermelo metric: 1 α g(v, v) + 1 α 2 g(w, v)2 1 g(w, v), α where α = 1 g(w, W ). Randers space forms are those Zermelo metrics having h of constant curvature and W a conformal Killing field M. A. Javaloyes (*) Flag Curvature 12 / 23
58 Flag constant curvature and stationary spacetimes Zermelo metric: 1 α g(v, v) + 1 α 2 g(w, v)2 1 g(w, v), α where α = 1 g(w, W ). Randers space forms are those Zermelo metrics having h of constant curvature and W a conformal Killing field Katok metrics are Randers space forms M. A. Javaloyes (*) Flag Curvature 12 / 23
59 Flag constant curvature and stationary spacetimes Zermelo metric: 1 α g(v, v) + 1 α 2 g(w, v)2 1 g(w, v), α where α = 1 g(w, W ). Randers space forms are those Zermelo metrics having h of constant curvature and W a conformal Killing field Katok metrics are Randers space forms When the Fermat metric associated to a stationary spacetime is of constant flag curvature, then the spacetime is locally conformally flat M. A. Javaloyes (*) Flag Curvature 12 / 23
60 Flag constant curvature and stationary spacetimes Zermelo metric: 1 α g(v, v) + 1 α 2 g(w, v)2 1 g(w, v), α where α = 1 g(w, W ). Randers space forms are those Zermelo metrics having h of constant curvature and W a conformal Killing field Katok metrics are Randers space forms When the Fermat metric associated to a stationary spacetime is of constant flag curvature, then the spacetime is locally conformally flat Reciprocal is not true ( h + df ) M. A. Javaloyes (*) Flag Curvature 12 / 23
61 Flag constant curvature and stationary spacetimes Zermelo metric: 1 α g(v, v) + 1 α 2 g(w, v)2 1 g(w, v), α where α = 1 g(w, W ). Randers space forms are those Zermelo metrics having h of constant curvature and W a conformal Killing field Katok metrics are Randers space forms When the Fermat metric associated to a stationary spacetime is of constant flag curvature, then the spacetime is locally conformally flat Reciprocal is not true ( h + df ) what about scalar flag curvature? M. A. Javaloyes (*) Flag Curvature 12 / 23
62 Schur s Lemma Theorem Let M be a Riemannian manifold with dimension 3. If for every point x M the sectional curvature does not depend on the on the plain, then M has constant sectional curvature. Issai Schur ( ) M. A. Javaloyes (*) Flag Curvature 13 / 23
63 Schur s Lemma Theorem Let M be a Riemannian manifold with dimension 3. If for every point x M the sectional curvature does not depend on the on the plain, then M has constant sectional curvature. It was established by Issai Schur ( ) Issai Schur ( ) M. A. Javaloyes (*) Flag Curvature 13 / 23
64 Schur s Lemma Theorem Let M be a Riemannian manifold with dimension 3. If for every point x M the sectional curvature does not depend on the on the plain, then M has constant sectional curvature. It was established by Issai Schur ( ) Issai Schur ( ) Generalized to Finsler manifolds by Lilia del Riego in her Phd. Thesis in M. A. Javaloyes (*) Flag Curvature 13 / 23
65 Gauss-Bonnet Theorem Theorem Suppose M is a 2-dim compact Riemannian manifold with boundary M. Then M K da + M k g ds = 2πχ(M), M. A. Javaloyes (*) Flag Curvature 14 / 23
66 Gauss-Bonnet Theorem Theorem Suppose M is a 2-dim compact Riemannian manifold with boundary M. Then M K da + M k g ds = 2πχ(M), Gauss knew a version but never published it Carl F. Gauss ( ) M. A. Javaloyes (*) Flag Curvature 14 / 23
67 Gauss-Bonnet Theorem Theorem Suppose M is a 2-dim compact Riemannian manifold with boundary M. Then M K da + M k g ds = 2πχ(M), Gauss knew a version but never published it Bonnet published a version in 1848 Pierre O. Bonnet ( ) M. A. Javaloyes (*) Flag Curvature 14 / 23
68 Gauss-Bonnet Theorem Theorem Suppose M is a 2-dim compact Riemannian manifold with boundary M. Then M K da + M k g ds = 2πχ(M), Gauss knew a version but never published it Bonnet published a version in 1848 Allendoerfer-Weil-Chern generalized Gauss-Bonnet to even dimensions using the Pfaffian in the mid-40 s S. S. Chern ( ) C. Allendoerfer ( ) André Weil ( ) M. A. Javaloyes (*) Flag Curvature 14 / 23
69 Gauss-Bonnet Theorem Theorem Suppose M is a 2-dim compact Riemannian manifold with boundary M. Then M K da + M k g ds = 2πχ(M), Gauss knew a version but never published it Bonnet published a version in 1848 Allendoerfer-Weil-Chern generalized Gauss-Bonnet to even dimensions using the Pfaffian in the mid-40 s Lichnerowitz (Comm. Helv. Math. 1949) extends the theorem to the Finsler setting in some particular cases André Lichnerowitz ( ) M. A. Javaloyes (*) Flag Curvature 14 / 23
70 Gauss-Bonnet Theorem Theorem Suppose M is a 2-dim compact Riemannian manifold with boundary M. Then M K da + M k g ds = 2πχ(M), Gauss knew a version but never published it Bonnet published a version in 1848 Allendoerfer-Weil-Chern generalized Gauss-Bonnet to even dimensions using the Pfaffian in the mid-40 s Lichnerowitz (Comm. Helv. Math. 1949) extends the theorem to the Finsler setting in some particular cases Bao-Chern (Ann. Math. 1996) extend it to a wider class of Finsler manifolds David Bao and S. S. Chern M. A. Javaloyes (*) Flag Curvature 14 / 23
71 Bonnet-Myers Theorem Theorem If Ricci curvature of a complete Riemannian manifold M is bounded below by (n 1)k > 0, then its diameter is at most π/ k and the manifold is compact. M. A. Javaloyes (*) Flag Curvature 15 / 23
72 Bonnet-Myers Theorem Theorem If Ricci curvature of a complete Riemannian manifold M is bounded below by (n 1)k > 0, then its diameter is at most π/ k and the manifold is compact. Pierre Ossian Bonnet ( ) obtained a version bounding from above the sectional curvatures with a positive constant Pierre O. Bonnet ( ) M. A. Javaloyes (*) Flag Curvature 15 / 23
73 Bonnet-Myers Theorem Theorem If Ricci curvature of a complete Riemannian manifold M is bounded below by (n 1)k > 0, then its diameter is at most π/ k and the manifold is compact. Pierre Ossian Bonnet ( ) obtained a version bounding from above the sectional curvatures with a positive constant Myers obtained the generalized version with Ric curvatures in 1941 M. A. Javaloyes (*) Flag Curvature 15 / 23
74 Bonnet-Myers Theorem Theorem If Ricci curvature of a complete Riemannian manifold M is bounded below by (n 1)k > 0, then its diameter is at most π/ k and the manifold is compact. Pierre Ossian Bonnet ( ) obtained a version bounding from above the sectional curvatures with a positive constant Myers obtained the generalized version with Ric curvatures in 1941 Louis Auslander extended the result to the Finsler setting in 1955 (Trans AMS) M. A. Javaloyes (*) Flag Curvature 15 / 23
75 Bonnet-Myers Theorem Theorem If Ricci curvature of a complete Riemannian manifold M is bounded below by (n 1)k > 0, then its diameter is at most π/ k and the manifold is compact. M. A. Javaloyes (*) Flag Curvature 16 / 23
76 Bonnet-Myers Theorem Theorem If Ricci curvature of a complete Riemannian manifold M is bounded below by (n 1)k > 0, then its diameter is at most π/ k and the manifold is compact. Bao-Chern-Chen assume just forward completeness in their book Introduction to Riemann-Finsler geometry D. Bao, S.S. Chern and Z. Shen M. A. Javaloyes (*) Flag Curvature 16 / 23
77 Bonnet-Myers Theorem Theorem If Ricci curvature of a complete Riemannian manifold M is bounded below by (n 1)k > 0, then its diameter is at most π/ k and the manifold is compact. Bao-Chern-Chen assume just forward completeness in their book Introduction to Riemann-Finsler geometry Causality reveals that completeness can be substituted by the condition B + (x, r) B (x, r) compact for all x M and r > 0 (see Caponio-M.A.J.-Sánchez, preprint 09) M. A. Javaloyes (*) Flag Curvature 16 / 23
78 Synge s Theorem Theorem If M is an even-dimensional, oriented, complete and connected manifold, with all the sectional curvatures bounded by some positive constant, then M is simply connected. M. A. Javaloyes (*) Flag Curvature 17 / 23
79 Synge s Theorem Theorem If M is an even-dimensional, oriented, complete and connected manifold, with all the sectional curvatures bounded by some positive constant, then M is simply connected. John Lighton Synge ( ) published this result in 1936 (Quaterly Journal of Mathematics). John Synge ( ) M. A. Javaloyes (*) Flag Curvature 17 / 23
80 Synge s Theorem Theorem If M is an even-dimensional, oriented, complete and connected manifold, with all the sectional curvatures bounded by some positive constant, then M is simply connected. John Lighton Synge ( ) published this result in 1936 (Quaterly Journal of Mathematics). Louis Auslander( ) extends the result for Finsler manifolds in 1955 John Synge ( ) M. A. Javaloyes (*) Flag Curvature 17 / 23
81 Synge s Theorem Theorem If M is an even-dimensional, oriented, complete and connected manifold, with all the sectional curvatures bounded by some positive constant, then M is simply connected. John Lighton Synge ( ) published this result in 1936 (Quaterly Journal of Mathematics). Louis Auslander( ) extends the result for Finsler manifolds in 1955 Again the completeness condition can be weakened. John Synge ( ) M. A. Javaloyes (*) Flag Curvature 17 / 23
82 Cartan-Hadamard Theorem Theorem If M is a geodesically complete connected Riemannian manifold of non positive sectional curvature. Then Geodesics do not have conjugate points exp p : T p M M is globally defined and a local diffeorphism If M simply connected, then exp p is a diffeomorphism M. A. Javaloyes (*) Flag Curvature 18 / 23
83 Cartan-Hadamard Theorem Theorem If M is a geodesically complete connected Riemannian manifold of non positive sectional curvature. Then Geodesics do not have conjugate points exp p : T p M M is globally defined and a local diffeorphism If M simply connected, then exp p is a diffeomorphism Jacques Hadamard ( ) Obtained for surfaces in 1898 by Hadamard Élie Cartan ( ) M. A. Javaloyes (*) Flag Curvature 18 / 23
84 Cartan-Hadamard Theorem Theorem If M is a geodesically complete connected Riemannian manifold of non positive sectional curvature. Then Geodesics do not have conjugate points exp p : T p M M is globally defined and a local diffeorphism If M simply connected, then exp p is a diffeomorphism Jacques Hadamard ( ) Obtained for surfaces in 1898 by Hadamard Generalized for every dimension by Cartan Élie Cartan ( ) M. A. Javaloyes (*) Flag Curvature 18 / 23
85 Cartan-Hadamard Theorem Theorem If M is a geodesically complete connected Riemannian manifold of non positive sectional curvature. Then Geodesics do not have conjugate points exp p : T p M M is globally defined and a local diffeorphism If M simply connected, then exp p is a diffeomorphism Obtained for surfaces in 1898 by Hadamard Generalized for every dimension by Cartan Extended to Finsler manifolds in 1955 by L. Auslander Jacques Hadamard ( ) Élie Cartan ( ) M. A. Javaloyes (*) Flag Curvature 18 / 23
86 Rauch s Comparison Theorem Theorem For large curvature, geodesics tend to converge, while for small (or negative) curvature, geodesics tend to spread. M. A. Javaloyes (*) Flag Curvature 19 / 23
87 Rauch s Comparison Theorem Theorem For large curvature, geodesics tend to converge, while for small (or negative) curvature, geodesics tend to spread. Proved in the 40 s by A. D. Aleksandrov for surfaces A. D. Aleksandrov ( ) M. A. Javaloyes (*) Flag Curvature 19 / 23
88 Rauch s Comparison Theorem Theorem For large curvature, geodesics tend to converge, while for small (or negative) curvature, geodesics tend to spread. Proved in the 40 s by A. D. Aleksandrov for surfaces Generalized to Riemannian manifolds in 1951 by H. E. Rauch A. D. Aleksandrov ( ) M. A. Javaloyes (*) Flag Curvature 19 / 23
89 Rauch s Comparison Theorem Theorem For large curvature, geodesics tend to converge, while for small (or negative) curvature, geodesics tend to spread. Proved in the 40 s by A. D. Aleksandrov for surfaces Generalized to Riemannian manifolds in 1951 by H. E. Rauch Probably P. Dazord was the first one in giving the generalized Rauch theorem in 1968 A. D. Aleksandrov ( ) M. A. Javaloyes (*) Flag Curvature 19 / 23
90 Sphere Theorem Theorem A simply connected connected manifold with < K 1 is homeomorphic to the sphere. 1 4 M. A. Javaloyes (*) Flag Curvature 20 / 23
91 Sphere Theorem Theorem A simply connected connected manifold with < K 1 is homeomorphic to the sphere. 1 4 Conjecture by Rauch. First proof by M. Berger in 1960 M. A. Javaloyes (*) Flag Curvature 20 / 23
92 Sphere Theorem Theorem A simply connected connected manifold with < K 1 is homeomorphic to the sphere. 1 4 Conjecture by Rauch. First proof by M. Berger in 1960 Alternative proof by Klingenberg in 1961 (obtaining homotopy equivalence rather than homeomorphism) M. A. Javaloyes (*) Flag Curvature 20 / 23
93 Sphere Theorem Theorem A simply connected connected manifold with < K 1 is homeomorphic to the sphere. 1 4 Conjecture by Rauch. First proof by M. Berger in 1960 Alternative proof by Klingenberg in 1961 (obtaining homotopy equivalence rather than homeomorphism) Dazord observes that Klingeberg proof works for reversible Finsler metrics in 1968 M. A. Javaloyes (*) Flag Curvature 20 / 23
94 Sphere theorem Theorem A simply connected connected manifold with < K 1 is homeomorphic to the sphere. 1 4 M. A. Javaloyes (*) Flag Curvature 21 / 23
95 Sphere theorem Theorem A simply connected connected manifold with < K 1 is homeomorphic to the sphere. 1 4 In 2004 H. B. Rademacher (Math. Ann.) extends Klingenberg proof to non-reversible Finsler metrics using the hypothesis ( λ) 2 < K 1), where λ = max{f ( X ) : F (X ) = 1} M. A. Javaloyes (*) Flag Curvature 21 / 23
96 Sphere theorem Theorem A simply connected connected manifold with < K 1 is homeomorphic to the sphere. 1 4 In 2004 H. B. Rademacher (Math. Ann.) extends Klingenberg proof to non-reversible Finsler metrics using the hypothesis ( λ) 2 < K 1), where λ = max{f ( X ) : F (X ) = 1} In 2007 S. Brendle and R. Schoen (J. Amer. Math. Soc 2009) prove by using Ricci-flow that there exists a diffeomorphism M. A. Javaloyes (*) Flag Curvature 21 / 23
97 Sphere theorem Theorem A simply connected connected manifold with < K 1 is homeomorphic to the sphere. 1 4 In 2004 H. B. Rademacher (Math. Ann.) extends Klingenberg proof to non-reversible Finsler metrics using the hypothesis ( λ) 2 < K 1), where λ = max{f ( X ) : F (X ) = 1} In 2007 S. Brendle and R. Schoen (J. Amer. Math. Soc 2009) prove by using Ricci-flow that there exists a diffeomorphism To obtain Rademacher s result it is enough symmetrized compact balls and bounded reversivility index M. A. Javaloyes (*) Flag Curvature 21 / 23
98 Inextendible theorems M. A. Javaloyes (*) Flag Curvature 22 / 23
99 Inextendible theorems Toponogov theorem? Problems with angles Victor A. Toponogov ( ) M. A. Javaloyes (*) Flag Curvature 22 / 23
100 Inextendible theorems Toponogov theorem? Problems with angles Submanifold theory (very difficult) Victor A. Toponogov ( ) M. A. Javaloyes (*) Flag Curvature 22 / 23
101 Inextendible theorems Toponogov theorem? Problems with angles Submanifold theory (very difficult) Laplacian theory Victor A. Toponogov ( ) M. A. Javaloyes (*) Flag Curvature 22 / 23
102 Bibliography D. Bao, S.-S. Chern, and Z. Shen, An introduction to Riemann-Finsler geometry, vol. 200 of Graduate Texts in Mathematics, Springer-Verlag, New York, M. A. Javaloyes (*) Flag Curvature 23 / 23
103 Bibliography D. Bao, S.-S. Chern, and Z. Shen, An introduction to Riemann-Finsler geometry, vol. 200 of Graduate Texts in Mathematics, Springer-Verlag, New York, M. A. Javaloyes (*) Flag Curvature 23 / 23
104 Bibliography D. Bao, S.-S. Chern, and Z. Shen, An introduction to Riemann-Finsler geometry, vol. 200 of Graduate Texts in Mathematics, Springer-Verlag, New York, M. A. Javaloyes (*) Flag Curvature 23 / 23
105 Bibliography D. Bao, S.-S. Chern, and Z. Shen, An introduction to Riemann-Finsler geometry, vol. 200 of Graduate Texts in Mathematics, Springer-Verlag, New York, M. A. Javaloyes (*) Flag Curvature 23 / 23
106 Bibliography D. Bao, S.-S. Chern, and Z. Shen, An introduction to Riemann-Finsler geometry, vol. 200 of Graduate Texts in Mathematics, Springer-Verlag, New York, M. A. Javaloyes (*) Flag Curvature 23 / 23
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