The transverse Jacobi equation Luis Guijarro Universidad Aut onoma - PowerPoint PPT Presentation

The transverse Jacobi equation Luis Guijarro Universidad Aut´ onoma de Madrid-ICMAT Symmetry and shape , 28th October 2019 Luis Guijarro (UAM-ICMAT) The transverse Jacobi equation October 28th, 2019 1 / 23

Summary The transverse Jacobi equation. 1 Comparison for the transverse Jacobi equation 2 Geometric applications. 3 Luis Guijarro (UAM-ICMAT) The transverse Jacobi equation October 28th, 2019 2 / 23

The linear symplectic space of Jacobi fields Joint work with Frederick Wilhelm (UCR). ( M n , g ) n -dimensional Riemannian manifold; γ : R → M unit speed geodesic. Definition Jac( γ ) denotes the 2( n − 1) -dimensional vector space of normal Jacobi fields along γ Jac( γ ) := { J Jacobi fields along γ : J , J ′ ⊥ γ ′ } . There is a linear symplectic form ω ( X , Y ) = � X ′ , Y � − � X , Y ′ � ω : Jac( γ ) × Jac( γ ) → R , Luis Guijarro (UAM-ICMAT) The transverse Jacobi equation October 28th, 2019 3 / 23

Lagrangian subspaces Definition Let W ⊂ Jac( γ ) a linear subspace. W is isotropic if ω | W × W ≡ 0 ; 1 L is Lagrangian if isotropic and maximal, i.e, dim L = ( n − 1) . 2 For a Lagrangian L , { J ( t ) : J ∈ L } = γ ′ ( t ) ⊥ , except at isolated points. Examples: Geodesic variations of γ leaving the initial point fixed; L 0 := { J : J (0) = 0 } Zeros: conjugate points. given N ⊂ M a submanifold, γ orthogonal to N at t = 0, J : J (0) ∈ T γ (0) N , J ′ (0) T + S γ ′ (0) J (0) = 0 � � L N := Zeros: focal points of N . Luis Guijarro (UAM-ICMAT) The transverse Jacobi equation October 28th, 2019 4 / 23

Vertical/horizontal bundles Choose W ⊂ L ; at each t ∈ R , let W ( t ) := { J ( t ) : J ∈ W } ⊕ { J ′ ( t ) : J ∈ W , J ( t ) = 0 } t → W ( t ) is a smooth bundle, inducing a smooth splitting γ ′ ( t ) ⊥ = W ( t ) ⊕ W ( t ) ⊥ = W ( t ) ⊕ H ( t ) . There is a covariant derivative of sections D ⊥ D ⊥ Y := Y ′ H dt : Γ( H ) → Γ( H ) , dt Luis Guijarro (UAM-ICMAT) The transverse Jacobi equation October 28th, 2019 5 / 23

Wilking’s O’Neill’s operators Definition Whenever possible, choose J ∈ W with J ( t ) = v, and define A t : W ( t ) → H ( t ) as A t ( v ) := J ′ H ( t ) , and A ∗ t : H ( t ) → W ( t ) as t ( v ) := J ′ W ( t ) . A ∗ A t , A ∗ t admit smooth extensions to all t . Luis Guijarro (UAM-ICMAT) The transverse Jacobi equation October 28th, 2019 6 / 23

The transverse Jacobi equation Theorem (Wilking, 2007) Let L ⊂ Jac( γ ) Lagrangian, and W ⊂ L. Then for any J ∈ L, we have that for every t ∈ R , D ⊥ 2 Y + { R ( t ) Y } H + 3 A t A ∗ t Y = 0 dt 2 where Y = J H . Luis Guijarro (UAM-ICMAT) The transverse Jacobi equation October 28th, 2019 7 / 23

Example: Riemannian submersions π : M n + k → B n Riemannian submersion. γ : R → M horizontal geodesic. Projectable Jacobi fields P : those arising from horizontally lifting to γ geodesic variations of ¯ γ := π ◦ γ in B . The holonomy vector fields are obtained lifting horizontally π ◦ γ to M : W = { J ∈ P : J vertical } ; dim W = k Any Lagrangian L ⊂ P contains W ; H are the horizontal parts. Wilking’s transverse equation for L / W ⇔ standard Jacobi equation for ¯ γ in B and A t is the O’Neill tensor. Luis Guijarro (UAM-ICMAT) The transverse Jacobi equation October 28th, 2019 8 / 23

Comparison Some overlap with results from Verdiani-Ziller. Definition (The Riccati operator) Define ˆ S ( t ) : H ( t ) → H ( t ) as ˆ S ( t )( v ) := J ′ ( t ) H , where J ∈ L with J ( t ) = v. Important: Well defined whenever any J ∈ L with J ( t ) = 0 lies in W . Definition W is of full index in an interval I ⊂ R if the above happens at every point of I. S t J H ′ = ( ˆ ′ + ˆ 2 ) J H S t J H ) ′ = ˆ ′ J H + ˆ ( ˆ S t S t S t Luis Guijarro (UAM-ICMAT) The transverse Jacobi equation October 28th, 2019 9 / 23

Wilking’s equation: ′ + ˆ 2 + { R ( t ) } H + 3 A t A ∗ ˆ S t S t t = 0 Taking traces leaves ′ + ˆ 2 + ˆ s t ˆ s t r t = 0 , where, if k = dim H , ˆ t the trace free part of ˆ S 0 S t , k Trace ˆ s t = 1 ˆ S t ; k ( | ˆ t | 2 + Trace R ( t ) H + 3 A t A ∗ r t = 1 S 0 ) ≥ 1 � � k Trace R ( t ) H ˆ t Intermediate Ricci curvature appears naturally: Definition Ric k ≥ ℓ if for any v ∈ T p M, and any ( k + 1) -orthogonal frame { v , e 1 , . . . , e k } we have k � sec( v , e i ) ≥ ℓ. i =1 Luis Guijarro (UAM-ICMAT) The transverse Jacobi equation October 28th, 2019 10 / 23

Riccati comparison, dimension one s ′ + s 2 + r ( t ) = 0 . (1) Denote by s a solutions of s ′ + s 2 + a = 0, with a = 1 , 0 , − 1. Lemma Suppose r ≥ a, and let s be a solution of (1) defined in [ t 0 , t max ] , with s ( t 0 ) ≤ s a ( t 0 ) . then s ( t ) ≤ s a ( t ) , if there is some t 1 with s ( t 1 ) = s a ( t 1 ) , then s ≡ s a and r ≡ a in [ t 0 , t 1 ] . Corollary If r ≥ 1 , [ t 0 , t max ] ⊂ [0 , π ] , and α ∈ [0 , π − t 0 ) , then the only solution of the Riccati equation with s ( t 0 ) ≤ cot ( t 0 + α ) that is defined in [ t 0 , π − α ) is s ( t ) = cot( t + α ) , and r ≡ 1 . Luis Guijarro (UAM-ICMAT) The transverse Jacobi equation October 28th, 2019 11 / 23

Positive intermediate Ricci comparison Theorem (F.Wilhelm, LG) Ric k ≥ k, and L a Lagrangian along γ with Riccati operator S t . If there is a k-dimensional subspace H 0 ⊥ γ ′ (0) such that the Ricatti operator for L satisfies Trace S 0 | H 0 ≤ 0 , then: There is some nonzero J ∈ L, J (0) ∈ H 0 , such that J ( t 1 ) = 0 for some 1 t 1 ∈ (0 , π/ 2] . If no J ∈ L vanishes before time π/ 2 , then there are subspaces W , H in L, 2 with H 0 = H 0 , such that L splits as L = W ⊕ H orthogonally for every t ∈ [0 , π/ 2] . Moreover, every field in H is of the form sin( t + π 2 ) · E ( t ) , where E is a parallel vector field. Luis Guijarro (UAM-ICMAT) The transverse Jacobi equation October 28th, 2019 12 / 23

Existence of focal points for Ric k . Theorem (F.Wilhelm, LG) Let M be a complete manifold with Ric k ≥ k, and N ⊂ M a submanifold (possibly not embedded, not complete) with dim N ≥ k. Then for any geodesic γ : R → M with γ (0) ∈ N, γ ′ (0) ⊥ N, there are at least 1 dim N − k + 1 focal points to N in the interval [ − π/ 2 , π/ 2] ; if for every geodesic as above, the first focal point is at time π/ 2 or − π/ 2 , 2 then N is totally geodesic Luis Guijarro (UAM-ICMAT) The transverse Jacobi equation October 28th, 2019 13 / 23

Diameter rigidity vs. focal rigidity Theorem (Gromoll-Grove’s diameter rigidity) Let M be a compact Riemannian manifold with sec ≥ 1 and diam = π/ 2 . Then M is homeomorphic to a sphere, or isometric to a compact projective space. Definition The focal radius of a submanifold N is the smallest time t 0 such that there is a focal point to N along a geodesic γ : R → M with γ (0) ∈ N, γ ′ (0) ⊥ N. Theorem (Focal rigidity, F. Wilhelm, LG) Let M be a compact Riemannian manifold with Ric k ≥ k; if M contains an embedded submanifold N with dim N ≥ k, and with focal radius π/ 2 , then the universal cover of M is isometric to a round sphere, or to a compact projective space with N totally geodesic in M. Luis Guijarro (UAM-ICMAT) The transverse Jacobi equation October 28th, 2019 14 / 23

Sphere Theorem for Ric k Definition Let γ : [0 , ℓ ] → M a unit geodesic. The index of γ is the number (with multiplicity) of conjugate points to γ (0) along γ . Lemma (Index of ”long” geodesics, F.Wilhelm, LG) ( M n , g ) with Ric k ≥ k. Then any unit geodesic γ : [0 , b ] → M with b ≥ π satisfies index( γ ) ≥ n − k . alez-´ Case of sec: David Gonz´ Alvaro, LG. Theorem (Sphere Theorem for Ric k ) ( M n , g ) with Ric k ≥ k. Suppose there is some p ∈ M with conj p > π/ 2 . Then the universal cover of M is ( n − k ) -connected; if k ≤ n / 2 , then the universal cover of M is homeomorphic to a sphere. . Luis Guijarro (UAM-ICMAT) The transverse Jacobi equation October 28th, 2019 15 / 23

Proof of the Sphere Theorem We can assume π 1 M = 0, n − k ≥ 2. Ω p : Loops based at p (or a finite dimensional manifold approximation). Enough to show that Ω p is ( n − k − 1)-connected . Lemma Let f : P → R be a proper, smooth function, and a < b such that every critical point in f − 1 [ a , b ] has index ≥ m. Then f − 1 ( −∞ , a ] ֒ → f − 1 ( −∞ , b ] is ( m − 1) -connected. E : Ω p → [0 , ∞ ) the energy function � ℓ E ( α ) = 1 | α ′ ( t ) | 2 dt 2 0 Its critical points are geodesic loops based at p . Luis Guijarro (UAM-ICMAT) The transverse Jacobi equation October 28th, 2019 16 / 23

Proof of the Sphere Theorem (cont.) Choose some π < ℓ < 2 conj p The long geodesic lemma ≡ geodesics longer than ℓ have index ≥ n − k , 1 ⇒ E − 1 ( −∞ , b / 2] ֒ → Ω p is ( n − k − 1)-connected. E − 1 ( −∞ , b / 2] has no critical points (contradiction). 2 n − k ≥ 2 and Ω p connected ⇒ E − 1 ( −∞ , b / 2] is connected. Any geodesic loop in E − 1 ( −∞ , b / 2] is not connected to { p } , because Lemma (Long homotopy lemma, Abresch-Meyer) γ : [0 , ℓ ] → M a unit geodesic loop based at p. If ℓ < 2 conj p , and { γ s } is a homotopy from γ to { p } , then there is some s 0 ∈ (0 , 1) such that length( γ s 0 ) > 2 conj p . Luis Guijarro (UAM-ICMAT) The transverse Jacobi equation October 28th, 2019 17 / 23

Thanks! Luis Guijarro (UAM-ICMAT) The transverse Jacobi equation October 28th, 2019 18 / 23