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On the Rauch Comparison Theorem and Its Applications

VIEWS: 66 PAGES: 13

Warm-up exercises (Warm-up), also known as ready to exercise, named after the former due to physiological responses, and the latter is a general concept. Warm-up exercises, the activities of some combination of the body; in the main physical activity prior to the lesser activity in advance of physical activity for more intense physical activity then prepare for the purpose of enhancing the efficiency of subsequent intense exercise, intense exercise Security, while meeting the physical and psychological human needs. Before exercise, the body's functional ability and efficiency can not be reached at the beginning of the highest level, and thus need to adjust the movement warm-up state.

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									                    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
                       2 sin 2t       3 sin 3t on [0, 3 ].
                                                      π
      Show that                   ≥




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
                 0.3                                                       sin3 t
                                                                       3



                 0.2


                 0.1


                                                                                            t
                                 0.2           0.4               0.6         0.8      1.0 tt


                             Figure:    Graphs of 1 sin 2t and 1 sin 3t
                                                  2            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

 Sturm's Theorem

      Theorem (Sturm)


      Let x1 (t ) and x2 (t ) be solutions to equations
                                       x1 (t ) + p1 (t )x1 (t ) = 0                                (1)


      and
                                       x2 (t ) + p2 (t )x2 (t ) = 0                                (2)

      respectively with initial conditions x1 (0) = x2 (0) = 0 and
      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



                                                                   x2 t




                  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
                    A Warm-up Exercise from Calculus         Some Ingredients
      Sturm's Theorem in Ordinary Dierential Equation       Rauch Comparison Theorem
        Rauch Comparison Theorem and Its Application         A Tool for the Proof
                                                             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 ), γ (t ))γ (t ) = 0,                             (3)
                               dt 2
      where     R   is the Riemann curvature tensor.




Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications
                    A Warm-up Exercise from Calculus     Some Ingredients
      Sturm's Theorem in Ordinary Dierential Equation   Rauch Comparison Theorem
        Rauch Comparison Theorem and Its Application     A Tool for the Proof
                                                         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
                    A Warm-up Exercise from Calculus     Some Ingredients
      Sturm's Theorem in Ordinary Dierential Equation   Rauch Comparison Theorem
        Rauch Comparison Theorem and Its Application     A Tool for the Proof
                                                         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
      |J1 (0)| = |J2 (0)| . Suppose that the sectional curvatures of M1 and
      M2 satisfy K1 ≤ K2 for all 2-planes containing γ1 and γ2 on each
      manifold. Then |J1 (t )| ≥ |J2 (t )| for all t ∈ [0, T ].

      Remark

      The condition normal Jacobi elds along                    γ1   and   γ2    can be
      replaced by the Jacobi elds satisfying
       J1 (0), γ1 (0) = J2 (0), γ2 (0)             , that we can see in the proof.

Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications
                    A Warm-up Exercise from Calculus     Some Ingredients
      Sturm's Theorem in Ordinary Dierential Equation   Rauch Comparison Theorem
        Rauch Comparison Theorem and Its Application     A Tool for the Proof
                                                         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
      |J1 (0)| = |J2 (0)| . Suppose that the sectional curvatures of M1 and
      M2 satisfy K1 ≤ K2 for all 2-planes containing γ1 and γ2 on each
      manifold. Then |J1 (t )| ≥ |J2 (t )| for all t ∈ [0, T ].

      Remark

      The condition normal Jacobi elds along                    γ1   and   γ2    can be
      replaced by the Jacobi elds satisfying
       J1 (0), γ1 (0) = J2 (0), γ2 (0)             , that we can see in the proof.

Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications
                    A Warm-up Exercise from Calculus     Some Ingredients
      Sturm's Theorem in Ordinary Dierential Equation   Rauch Comparison Theorem
        Rauch Comparison Theorem and Its Application     A Tool for the Proof
                                                         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


                                        ˆt
                       It (V , V ) :=        ( V ,V      − R (γ , V )γ , V )dt .                   (4)

                                         0




Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications
                    A Warm-up Exercise from Calculus     Some Ingredients
      Sturm's Theorem in Ordinary Dierential Equation   Rauch Comparison Theorem
        Rauch Comparison Theorem and Its Application     A Tool for the Proof
                                                         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 J , γ = 0,
      and V be a piecewise dierentiable vector eld along γ , with
       V , γ = 0. Suppose that J (0) = V (0) = 0 and J (t0 ) = V (t0 ),
      t0 ∈ (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
                    A Warm-up Exercise from Calculus     Some Ingredients
      Sturm's Theorem in Ordinary Dierential Equation   Rauch Comparison Theorem
        Rauch Comparison Theorem and Its Application     A Tool for the Proof
                                                         Application of Rauch Comparison Theorem

 Sphere Theorem




      Theorem (Berger-Klingenberg, 1960)

      Any compact, simply connected, and strictly 1 -pinched manifold
                                                     4
      M n , that is the sectional curvature K of M n satisfying
            1
      0 < Kmax < K ≤ Kmax , is homeomorphic to S .
                                                        n
            4




Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications
                    A Warm-up Exercise from Calculus     Some Ingredients
      Sturm's Theorem in Ordinary Dierential Equation   Rauch Comparison Theorem
        Rauch Comparison Theorem and Its Application     A Tool for the Proof
                                                         Application of Rauch Comparison Theorem

 Dierentiable Sphere Theorem




      Theorem (Brendle and Schoen, 2009)

      Any compact, simply connected and strictly 1 -strictly pinched
                                                 4
      manifold M n is dieomorphic to S n .




Yang Liu, Riemannian and Sub-Riemannian Geometry Seminar On the Rauch Comparison Theorem and Its Applications

								
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