A Hopf Theorem and Related Results for Pseduo-Riemannian Geometry 公开

Abstract

We show that a complete Pseudo-Riemannian metric without conjugate points along time-like curves which is flat outside of a compact set must be flat inside that compact set. This type of result is called a Hopf theorem, and our result is a generalization of a result by Croke in the Riemannian case. We use a mixture of geometric methods from that work and methods used in showing boundary rigidity through integral geometry. During the course of this three part proof we show other related results, including that the geodesic ray transform of functions over time-like curves for separable Pseudo- Riemannian manifolds is injective.

Table of Contents

Contents

1 Introduction and Main Results 1

1.1 The theorem of E.Hopf ... 1

1.2 Main results ... 2

1.3 Outline of the proof ... 4

1.4 Historical results ... 5

1.4.1 Historical Hopf-type theorems ... 5

1.4.2 Historical reductions to geodesic ray transforms ... 7

1.4.3 Historical injectivity for geodesic ray transforms ... 7

2 Reduction to Geometric Inverse Problems 10

2.1 Flat on a complement set in Riemannian geometry ... 10 

2.2 Flat on a compliment set in Pseduo-Riemannian geometry ... 17

3 From Boundary Rigidity to Integral Geometry 28

3.1 Formulation of the problem ... 28 

3.2 Derivation of the geodesic ray transform ... 30 

3.3 Proof of the Hopf-type theorem ... 34 

3.4 Remarks on tensor problems ... 35

4 Analysis of Geodesic Ray Transforms 38

4.1 Basics of pseudo-differential operators ... 40

4.2 The normal operator for the flat metric ... 43 

4.3 The normal operator for general metrics ... 47 

4.4 The generic injectivity and stability ... 51

Bibliography 54

About this Dissertation

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School	Laney Graduate School
Department	Mathematics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Mathematics
关键词	differential geometry pseudo-riemannian geometry pseudo-differential operators integral operators
Committee Chair / Thesis Advisor	Yiran Wang, Emory University
Committee Members	David Borthwick, Emory University Maja Taskovic, Emory University

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