Maximal Diameter Sphere Theorem for Manifolds with Nonconstant Radial Curvature

Maximal diameter sphere theorem ( 最大直径球面定理 )

Nathaphon BoonnamSc.D. student from Department of Mathematics

School of Science and TechnologyTokai University

Outline

• Introduction• Preliminaries• Proof of Main Theorem

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Outline


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Introduction

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Introduction

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Introduction

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Introduction

A natural extension of the maximal diameter sphere theorem by the radial curvature would be;

For a complete connected Riemannian manifold M whose radial sectional curvature at a point p bounded from below by 1,

• is the diameter of M at most ?

• Furthermore, if the diameter of M equals , then

is the manifold M isometric to the unit sphere ?

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Introduction

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Here we define the radial plane and radial curvature from a point p of a complete connected Riemannian manifold M.

• For each point q ( ), 2-dimensional linear subspace of is called a radial plane at q if a unit speed minimal geodesic segment satisfying

• the sectional curvature of is called a radial curvature at p.

Introduction

• Is the diameter of M at most ?

This problem can be affirmatively solved. It is an easy con-sequence from Generalized Toponogov Comparison Theorem

• Furthermore, if the diameter of M equals , then

is the manifold M isometric to the unit sphere?

This problem is still open!! But one can generalize the maximal diameter sphere theorem for a manifold which has radial curvature at a point p bounded from below by the radial curvature function of a 2-sphere of revolution if the 2-sphere of revolution belongs to a certain class.

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Introduction

: a complete Riemannian manifold homeomorphic to 2-sphere

is called a 2-sphere of revolution if admits a point such that for any two points on with

, an isometry f on satisfying

The point is called a pole of

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MM M

M

M

M

f

Introduction

It is proved in [ST] that has another pole ,

The Riemannian metric g of is expressed as

on ,

where : geodesic polar coordinates

and

Hence has a pair of poles and .

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MM

M

• : a pole of and fix it.Each unit speed geodesic emanating fromis called a meridian.• It is observed in [ST] that each meridian

where passes through and is periodic.• The function is

called the radial curvature function of ,where G : the Gaussian curvature of .

MM

M

Introduction

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Introduction

A 2-sphere of revolution with a pair of poles and is called a model surface if satisfies the following two properties :

(1.1) has a reflective symmetry with respect to the equator,

.

(1.2) The Gaussian curvature G of is strictly decreasing along a meridian from a point to the point on equator.

MM

M

M

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Introduction

A typical example of a model surface is an ellipsoid of prolate type, i.e., the surface defined by

The point are a pair of poles

and is the equator.

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Introduction

The surface generated by the plane curve

is a model surface, where

It is easy to see that the Gaussian curvature of

the equator is –1.

Figure from [ST]

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Introduction

For each 2-dimensional model with a Riemannian metric

,

we define an n-dimensional model homeomorphic to an

n-sphere with a Riemannian metric

where : the Riemannian metric of the

dimensional unit sphere

M

nM

nS

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For example, an ellipsoid of prolate type defined by

We get the n-dimensional model of the ellipsoid above is the n-dimensional ellipsoid defined by

.

Introduction

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Introduction

M : a complete n-dimensional Riemannian manifold with a base point p.

M is said to have radial sectional curvature at p bounded from below by the radial curvature function of a model surface if for any point and any radial plane at q, the sectional curvature of M satisfies

M

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Introduction

In this study, we generalize the maximal diameter sphere theorem as follows:

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Introduction

As a corollary, we get an interesting result:

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Outline


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Preliminaries

Review the notion of a cut point and a cut locus

M : a complete Riemannian manifold with a base point p.

: a unit speed minimal geodesic segment

emanating from on M.

If any extended geodesic segment of ,

b > a, is not minimizing arc joining p to anymore,

then the endpoint of the geodesic segment is called

a cut point of p along .

Not minimal!!

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Preliminaries

For each point p on M, the cut locus is defined by the set of all cut points along the minimal geodesic segments emanating from p.

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Preliminaries

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Preliminaries

The cut locus by Klingenberg Lemma

25Remark: It is known that the cut locus has a local tree structure for 2-dimensional Riemannian manifold.

Preliminaries

We need the following two theorems, which was proved by Sinclair and Tanaka [ST].

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Preliminaries

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Preliminaries

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Preliminaries

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小大

大

大小

小

Preliminaries

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Preliminaries

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(By Leibniz rule)

Preliminaries

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Preliminaries

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Preliminaries

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Preliminaries

Remark that if there does not exist a conjugate point of

along , then, by the Rauch comparison theorem, there does not exist a conjugate point of along .

More curved surfaces, conjugate points appear faster than on less curved manifolds.

Rauch

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Outline


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Proof of Main Theorem

M : a complete Riemannian manifold with a base point p

: a 2-sphere of revolution with a pair of poles satisfying

(1.1) and (1.2) in the introduction.• From now on, we assume that M has radial sectional curvature

at p bounded from below by the radial curvature function of .

• By scaling the Riemannian metrics of M and ,

we may assume that

M

MM

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From Lemma 3.1,

by Theorem 2.2;,

.


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By the triangle inequality,

and Lemma 3.1,

we get,

,

,

.


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By the triangle inequality,

and Lemma 3.2,

we get,

and it is easy to see that q is a unique cut point of p because .


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References

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References

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Maximal diameter sphere theorem( 最大直径球面定理 )

Nathaphon BoonnamSc.D. student from Department of Mathematics

School of Science and TechnologyTokai University