Rotationally Symmetric Planes in Comparison Geometry Público

Abstract

Kondo-Tanaka generalized the Toponogov Comparison Theorem so that an arbitrary noncompact manifold M can be compared with a rotationally symmetric plane M_m, and they used this to show that if M_m satisfies certain conditions, then M must be topologically finite. We substitute one of the conditions for M_m with a weaker condition and show that our method using this weaker condition enables us to draw further conclusions on the topology of M. We also completely remove one of the conditions required for the Sector Theorem, another important result by Kondo-Tanaka. Cheeger-Gromoll showed that if M has nonnegative sectional curvature, then M contains a boundaryless, totally convex, compact submanifold S, called a soul, such that M is homeomorphic to the normal bundle over S. We show that in the case of a rotationally symmetric plane M_m, the set of souls is a closed geometric ball centered at the origin, and if furthermore M_m is von Mangoldt, then the radius of this ball can be explicitly determined. We prove that the set of critical points of infinity in M_m is equal to this set of souls, and we make observations on the set of critical points of infinity when M_m is von Mangoldt with negative sectional curvature near infinity. Finally we set out conditions under which M_m can be guaranteed an annulus free of critical points of infinity and show that we can construct a von Mangoldt plane M_m that is a cone near infinity and for which the slope near infinity is prescribed to be any number in (0, 1].

Table of Contents

1 Introduction 3 1.1 Short Introduction . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Long Introduction . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . 14 2 Basic Facts and Denitions 16 2.1 Notations and Conventions . . . . . . . . . . . . . . . . . . 19 2.2 Sectional Curvature . . . . . . . . . . . . . . . . . . . . . . 20 2.3 The Gauss-Bonnet Theorem . . . . . . . . . . . . . . . . . 21 2.4 Rotationally Symmetric Planes . . . . . . . . . . . . . . . 22 2.5 The Cut Locus in a von Mangoldt Plane . . . . . . . . . . 23 2.5.1 Conjugate and Focal Points . . . . . . . . . . . . . 23 2.5.2 The Sturm Comparison Theorem . . . . . . . . . . 24 2.5.3 The Structure of a Cut Locus in a Rotationally Symmetric Plane . . . . . . . . . . . . . . . . . . . . . . 25 3 The Soul Theorem 31 4 Geodesics and Rays 38 4.1 The Clairaut Relation . . . . . . . . . . . . . . . . . . . . 38 4.2 The Turn Angle Formula . . . . . . . . . . . . . . . . . . . 39 4.3 Various lemmas and theorems . . . . . . . . . . . . . . . . 41 4.4 Planes of Nonnegative Curvature . . . . . . . . . . . . . . 54 5 Critical Points of Innity in a Rotationally Symmetric Plane 56 5.1 Critical Points of Innity when Curvature is Nonnegative . 56 5.2 Critical Points of Innity and Poles . . . . . . . . . . . . . 59 5.3 Critical Points of Innity in a von Mangoldt Plane with Negative Curvature . . . . . . . . . . . . . . . . . . . . . . 61 5.4 Creating Annuli Free of Critical Points of Innity . . . . . 63 6 Souls in a Rotationally Symmetric Plane 66 7 More on von Mangoldt Planes 72 7.1 Some Observations . . . . . . . . . . . . . . . . . . . . . . 72 7.2 Smoothed cones made von Mangoldt . . . . . . . . . . . . 76 8 Extending the Work of Kondo and Tanaka 79 8.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 8.2 The Generalized Toponogov Comparison Theorem . . . . . 81 8.3 The Two Theorems . . . . . . . . . . . . . . . . . . . . . . 83 8.4 Extending the Main Theorem . . . . . . . . . . . . . . . . 83 8.5 Improving on the Sector Theorem . . . . . . . . . . . . . . 89

Bibliography 97

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School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Mathematics
Palavra-chave	critical point surface of revolution topological finiteness von Mangoldt soul Toponogov
Committee Chair / Thesis Advisor	Belegradek, Igor, Georgia Tech
Committee Members	Borthwick, David, Emory University Abrams, Aaron, Emory University

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