# On the Rauch Comparison Theorem and Its Applications

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```					                    A Warm-up Exercise from Calculus
Sturm's Theorem in Ordinary Dierential 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 Dierential 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 Dierential 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 Dierential 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 Dierential 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 Dierential Equation       Rauch Comparison Theorem
Rauch Comparison Theorem and Its Application         A Tool for the Proof
Application of Rauch Comparison Theorem

Jacobi Field

Denition (Jacobi Field)

A vector eld        J   along a geodesic          g   in a Riemmannian manifold is
said to be a Jacobi eld if it satises 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 Dierential Equation   Rauch Comparison Theorem
Rauch Comparison Theorem and Its Application     A Tool for the Proof
Application of Rauch Comparison Theorem

Conjugate Point

Denition (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 Dierential 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 Dierential 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 Dierential Equation   Rauch Comparison Theorem
Rauch Comparison Theorem and Its Application     A Tool for the Proof
Application of Rauch Comparison Theorem

Index Form

Denition (Index Form)

The index form of a piecewise dierentiable vector eld                         V   along a
geodesic     γ   on a Riemannian manifold              M   is dened 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 Dierential 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 dierentiable 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 Dierential 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 Dierential Equation   Rauch Comparison Theorem
Rauch Comparison Theorem and Its Application     A Tool for the Proof
Application of Rauch Comparison Theorem

Dierentiable Sphere Theorem

Theorem (Brendle and Schoen, 2009)

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

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

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Description: 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.