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TABLE OF CONTENTS

Preface

Chapter I. METRIC SPACES WITH GEODESICS

§1. Metric Spaces; Notations

§2. The Basic Axioms

§3. Geodesics

§4. Topological Structure of One- and Two-dimensional Spaces With Axioms A - D

Chapter II. METRIC CONDITIONS FOR FINSLER SPACES

§1. Convex Surfaces and Minkowski Metrics

§2. Riemann Spaces and Finsler Spaces

§3. Condition ∆(P) and the Definition of the Local Metric

§4. Equivalence of the Local Metric with the Original Metric, and its Convexity

§5. The Minkowskian Character of the Local Metric

§6. The Continuity of the Local Metric

Chapter III. PROPERTIES OF GENERAL S. L. SPACES (Spaces with a unique geodesic through any two points)

§1. Axiom E. Shape of the Geodesics

§2. Two Dimensional S. L. Spaces

§3. The Inverse Problem for the Euclidean Plane

§4. Asymptotes and Limit Spheres

§5. Examples on Asymptotes and Limit Spheres. The Parallel Axioms

§6. Desarguesian Spaces

Chapter IV. SPACES WITH CONVEX SPHERES

§1. The Convexity Condition

§2. Characterization of the Higher Dimensional Elliptic Geometry

§3. Perpendiculars in Spaces with Spheres of Order 2

§4. Perpendiculars and Baselines in Open S. L. Spaces

§5. Definition and Properties of Limit Bisectors

§6. Characterizations of the Higher Dimensional Minkowskian and Euclidean Geometries

§7. Plane Minkowskian Geometries

§8. Characterization of Absolute Geometry

Chapter V. MOTIONS

§1. Definition of Motions. Involutoric Motions in S. L. Spaces

§2. Free Movability

§3. Example of a Non-homogeneous Riemann Space in which Congruent Pairs of Points Can be Moved into Each Other

§4. Translations Along g and the Asymptotes to g

§5. Quasi-hyperbolic Metrics

§6. Translations Along Non-parallel Lines and in Closed Planes

§7. Plane Geometries with a Transitive Group of Motions

§8. Transitive Abelian Groups of Motions in Higher Dimensional Spaces

§9. Some Problems Regarding S. L. Spaces and Other Spaces

Literature

Index