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Chapter Ⅰ．METRIC SPACES WITH GEODESICS 1

1．Metric Spaces；Notations 1

2．The Basic Axioms 11

3．Geodesics 17

4．Topological Structure of One- and Two- dimensional Spaces With Axioms A - D 24

Chapter Ⅱ．METRIC CONDITIONS FOR FINSLER SPACES 30

1．Convex Surfaces and Minkowski Metrics 31

2．Riemann Spaces and Finsler Spaces 40

3．Condition Δ(P) and the Definition of the Local Metric 47

4．Equivalence of the Local Metric with the Original Metric，and its Convexity 53

5．The Minkowskian Character of the Local Metric 57

6．The Continuity of the Local Metric 63

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

1．Axiom E．Shape of the Geodesics 73

2．Two Dimensional S．L．Spaces 79

3．The Inverse Problem for the Euclidean Plane 89

4．Asymptotes and Limit Spheres 98

5．Examples on Asymptotes and Limit Spheres The Parallel Axioms 105

6．Desarguesian Spaces 113

Chapter Ⅳ．SPACES WITH CONVEX SPHERES 119

1．The Convexity Condition 120

2．Characterization of the Higher Dimensional Elliptic Geometry 124

3．Perpendiculars in Spaces with Spheres of Order 2 132

4．Perpendiculars and Baselines in Open S．L．Spaces 139

5．Definition and Properties of Limit Bisectors 146

6．Characterizations of the Higher Dimensional Minkowskian and Euclidean Geometries 154

7．Plane Minkowskian Geometries 160

8．Characterization of Absolute Geometry 168

Chapter Ⅴ．MOTIONS 175

1．Definition of Motions．Involutoric Motions in S．L．Spaces 176

2．Free Movability 184

3．Example of a Non-homogeneous Riemann Space in which Congruent Pairs of Points Can be Moved into Each Other 192

4．Translations Along g and the Asymptotes to g 198

5．Quasi-hyperbolic Metrics 208

6．Translations Along Non-parallel Lines and in Closed Planes 214

7．Plane Geometries with a Transitive Group of Motions 220

8．Transitive Abelian Groups of Motions in Higher Dimensional Spaces 228

9．Some Problems Regarding S．L．Spaces and Other Spaces 232

Literature 235

Index 240