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On the Rauch Comparison Theorem and Its Applications A Warm-up Exercise from Calculus Sturm's Theorem in Ordinary Dierential Equation Rauch Comparison Theorem and Its Application On the Rauch Comparison Theorem and Its Applications Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar April 30, 2009 Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications A Warm-up Exercise from Calculus Sturm's Theorem in Ordinary Dierential Equation Rauch Comparison Theorem and Its Application A Warm-up Exercise from Calculus Exercise 1 1 π Show that 2 sin 2t ≥ 3 sin 3t on [0; 3 ]. Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications A Warm-up Exercise from Calculus Sturm's Theorem in Ordinary Dierential Equation Rauch Comparison Theorem and Its Application A Warm-up Exercise from Calculus 0.5 1 sin2 t 2 0.4 1 sin3 t 0.3 3 0.2 0.1 t 0.2 0.4 0.6 0.8 1.0 tt 1 1 Figure: Graphs of 2 sin 2t and 3 sin 3t Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications A Warm-up Exercise from Calculus Sturm's Theorem in Ordinary Dierential Equation Rauch Comparison Theorem and Its Application Sturm's Theorem Theorem (Sturm) Let x1(t) and x2(t) be solutions to equations 00 x1 (t) + p1(t)x1(t) = 0 (1) and 00 x2 (t) + p2(t)x2(t) = 0 (2) respectively with initial conditions x1(0) = x2(0) = 0 and 0 0 x1(0) = x2(0) = 1, where p1(t) and p2(t) are continuous on [0; T ]. Suppose p1(t) ≤ p2(t) on [0; T ] and x2(t) > 0 on (0; T ]. Then x1(t) ≥ x2(t) on [0; T ]. Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications A Warm-up Exercise from Calculus Sturm's Theorem in Ordinary Dierential Equation Rauch Comparison Theorem and Its Application Sturm's Theorem x1 t H L x2 t H L 0 T t Figure: Comparison of the Solutions to Two ODEs Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications Some Ingredients A Warm-up Exercise from Calculus Rauch Comparison Theorem Sturm's Theorem in Ordinary Dierential Equation A Tool for the Proof Rauch Comparison Theorem and Its Application Application of Rauch Comparison Theorem Jacobi Field Denition (Jacobi Field) A vector eld J along a geodesic g in a Riemmannian manifold is said to be a Jacobi eld if it satises the Jacobi equation D2 J t R J t 0 t 0 t 0 (3) dt2 ( ) + ( ( ); γ ( ))γ ( ) = ; where R is the Riemann curvature tensor. Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications Some Ingredients A Warm-up Exercise from Calculus Rauch Comparison Theorem Sturm's Theorem in Ordinary Dierential Equation A Tool for the Proof Rauch Comparison Theorem and Its Application Application of Rauch Comparison Theorem Conjugate Point Denition (Conjugate Point) A point q is said to be a conjugate point to another point p along a geodesic g in a Riemannian manifold if there exists a non-zero Jacobi eld along g that vanishes at p and q. Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications Some Ingredients A Warm-up Exercise from Calculus Rauch Comparison Theorem Sturm's Theorem in Ordinary Dierential Equation A Tool for the Proof Rauch Comparison Theorem and Its Application Application of Rauch Comparison Theorem The Rauch Comparison Theorem Theorem (Rauch, 1951) Let M1 and M2 be Riemannian manifolds, γ1 :[0; T ] ! M1 and γ2 :[0; T ] ! M2 be normalized geodesic segments such that γ2(0) has no conjugate points along γ2, and J1,J2 be normal Jacobi elds along γ1 and γ2 such that J1(0) = J2(0) = 0 and 0 0 jJ1(0)j = jJ2(0)j . Suppose that the sectional curvatures of M1 and 0 0 M2 satisfy K1 ≤ K2 for all 2-planes containing γ1 and γ2 on each manifold. Then jJ1(t)j ≥ jJ2(t)j for all t 2 [0; T ]. Remark The condition normal Jacobi elds along γ1 and γ2 can be replaced by the Jacobi elds satisfying 0 0 0 0 hJ1(0); γ1(0)i = hJ2(0); γ2(0)i, that we can see in the proof. Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications Some Ingredients A Warm-up Exercise from Calculus Rauch Comparison Theorem Sturm's Theorem in Ordinary Dierential Equation A Tool for the Proof Rauch Comparison Theorem and Its Application Application of Rauch Comparison Theorem The Rauch Comparison Theorem Theorem (Rauch, 1951) Let M1 and M2 be Riemannian manifolds, γ1 :[0; T ] ! M1 and γ2 :[0; T ] ! M2 be normalized geodesic segments such that γ2(0) has no conjugate points along γ2, and J1,J2 be normal Jacobi elds along γ1 and γ2 such that J1(0) = J2(0) = 0 and 0 0 jJ1(0)j = jJ2(0)j . Suppose that the sectional curvatures of M1 and 0 0 M2 satisfy K1 ≤ K2 for all 2-planes containing γ1 and γ2 on each manifold. Then jJ1(t)j ≥ jJ2(t)j for all t 2 [0; T ]. Remark The condition normal Jacobi elds along γ1 and γ2 can be replaced by the Jacobi elds satisfying 0 0 0 0 hJ1(0); γ1(0)i = hJ2(0); γ2(0)i, that we can see in the proof. Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications Some Ingredients A Warm-up Exercise from Calculus Rauch Comparison Theorem Sturm's Theorem in Ordinary Dierential Equation A Tool for the Proof Rauch Comparison Theorem and Its Application Application of Rauch Comparison Theorem Index Form Denition (Index Form) The index form of a piecewise dierentiable vector eld V along a geodesic γ on a Riemannian manifold M is dened as t 0 0 0 0 It (V ; V ) := (hV ; V i − hR(γ ; V )γ ; V i)dt: (4) ˆ 0 Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications Some Ingredients A Warm-up Exercise from Calculus Rauch Comparison Theorem Sturm's Theorem in Ordinary Dierential Equation A Tool for the Proof Rauch Comparison Theorem and Its Application Application of Rauch Comparison Theorem Index Lemma Lemma (Index Lemma) Let J be a Jacobi eld along a geodesic γ :[0; T ] ! M, which has no conjugate point to γ(0) in the interval (0; T ], with hJ; γ0i = 0, and V be a piecewise dierentiable vector eld along γ, with 0 hV ; γ i = 0. Suppose that J(0) = V (0) = 0 and J(t0) = V (t0), t0 2 (0; T ]. Then It0 (J; J) ≤ It0 (V ; V ). Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications Some Ingredients A Warm-up Exercise from Calculus Rauch Comparison Theorem Sturm's Theorem in Ordinary Dierential Equation A Tool for the Proof Rauch Comparison Theorem and Its Application Application of Rauch Comparison Theorem Sphere Theorem Theorem (Berger-Klingenberg, 1960) 1 Any compact, simply connected, and strictly 4 -pinched manifold Mn, that is the sectional curvature K of Mn satisfying 1 n 0 < 4 Kmax < K ≤ Kmax , is homeomorphic to S . Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications Some Ingredients A Warm-up Exercise from Calculus Rauch Comparison Theorem Sturm's Theorem in Ordinary Dierential Equation A Tool for the Proof Rauch Comparison Theorem and Its Application Application of Rauch Comparison Theorem Dierentiable Sphere Theorem Theorem (Brendle and Schoen, 2009) 1 Any compact, simply connected and strictly 4 -strictly pinched manifold Mn is dieomorphic to Sn. Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications.
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