《Metric Methods in Finsler Spaces and in The Foundations of Geometry》

Chapter Ⅰ．METRIC SPACES WITH GEODESICS1

1．Metric Spaces；Notations1

2．The Basic Axioms11

3．Geodesics17

4．Topological Structure of One- and Two- dimensional Spaces With Axioms A - D24

Chapter Ⅱ．METRIC CONDITIONS FOR FINSLER SPACES30

1．Convex Surfaces and Minkowski Metrics31

2．Riemann Spaces and Finsler Spaces40

3．Condition Δ(P) and the Definition of the Local Metric47

4．Equivalence of the Local Metric with the Original Metric，and its Convexity53

5．The Minkowskian Character of the Local Metric57

6．The Continuity of the Local Metric63

Chapter Ⅲ．PROPERTIES OF GENERAL S．L．SPACES(Spaces with a unique geodesic through any two points)72

1．Axiom E．Shape of the Geodesics73

2．Two Dimensional S．L．Spaces79

3．The Inverse Problem for the Euclidean Plane89

4．Asymptotes and Limit Spheres98

5．Examples on Asymptotes and Limit Spheres The Parallel Axioms105

6．Desarguesian Spaces113

Chapter Ⅳ．SPACES WITH CONVEX SPHERES119

1．The Convexity Condition120

2．Characterization of the Higher Dimensional Elliptic Geometry124

3．Perpendiculars in Spaces with Spheres of Order 2132

4．Perpendiculars and Baselines in Open S．L．Spaces139

5．Definition and Properties of Limit Bisectors146

6．Characterizations of the Higher Dimensional Minkowskian and Euclidean Geometries154

7．Plane Minkowskian Geometries160

8．Characterization of Absolute Geometry168

Chapter Ⅴ．MOTIONS175

1．Definition of Motions．Involutoric Motions in S．L．Spaces176

2．Free Movability184

3．Example of a Non-homogeneous Riemann Space in which Congruent Pairs of Points Can be Moved into Each Other192

4．Translations Along g and the Asymptotes to g198

5．Quasi-hyperbolic Metrics208

6．Translations Along Non-parallel Lines and in Closed Planes214

7．Plane Geometries with a Transitive Group of Motions220

8．Transitive Abelian Groups of Motions in Higher Dimensional Spaces228

9．Some Problems Regarding S．L．Spaces and Other Spaces232

Literature235

Index240