INTRINSIC GEOMETRY OF SURFACES
CONTENTS CHAPTER I BASIC CONCEPTS AND RESULTS 1 THE GENERAL CONCEPT AND PROBLEMS OF INTRINSIC GEOMETRY 2 GAUSSIAN INTRINSIC GEOMETRY 3 A POLYHEDRAL METRIC 4 DEVELOPMENT 5 PASSAGE FROM POLYHEDRA TO ARBITRARY SURFACES 6 A MANIFOLD WITH AN INTRINSIC METRIC 7 BASIC CONCEPTS OF INTRINSIC GEOMETRY 8 CURVATURE 9 CHARACTERISTIC PROPERTIES OF THE INTRINSIC METRIC 10 SOME SINGULARITIES OF THE INTRINSIC GEOMETRY OF CONVEX SURFACES 11 THEOREMS OF THE INTRINSIC GEOMETRY OF CONVEX SURFACES CHAPTER II GENERAL PROPOSITIONS ABOUT INTRINSIC METRICS 1 GENERAL THEOREMS ON RECTI?ABLE CURVES 2 GENERAL THEOREMS ON SHORTEST ARCS 3 THE NONOVERLAPPING CONDITION FOR SHORTEST ARCS 4 A CONVEX NEIGHBORHOOD 5 GENERAL PROPERTIES OF CONVEX DOMAINS 6 TRIANGULATION CHAPTER III CHARACTERISTIC PROPERTIES OF THE INTRINSIC METRIC 1 CONVERGENCE OF THE METRICS OF CONVERGENT CONVEX SURFACES 2 THE CONVEXITY CONDITION FOR A POLYHEDRAL METRIC 3 THE CONVEXITY CONDITION FOR THE METRIC OF A CONVEX SURFACE 4 CONSEQUENCES OF THE CONVEXITY CONDITION CHAPTER IV THE ANGLE 1 GENERAL THEOREMS ON ADDITION OF ANGLES 2 THEOREMS ON ADDITION OF ANGLES ON CONVEX SURFACES 3 THE ANGLE OF A SECTOR BOUNDED BY SHORTEST ARCS 4 ON CONVERGENCE OF ANGLES 5 THE TANGENT CONE 6 THE SPATIAL MEANING OF THE ANGLE BETWEEN SHORTEST ARCS CHAPTER V CURVATURE 1 INTRINSIC CURVATURE 2 THE AREA OF A SPHERICAL IMAGE 3 GENERALIZATION OF THE GAUSS THEOREM 4 THE CURVATURE OF A BOREL SET 5 DIRECTIONS IN WHICH IT IS IMPOSSIBLE TO DRAW A SHORTEST ARC 6 CURVATURE AS MEASURE OF NON-EUCLIDICITY OF THE METRIC CHAPTER VI EXISTENCE OF A CONVEX POLYHEDRON WITH A GIVEN METRIC 1 ON DETERMINING A METRIC FROM A DEVELOPMENT 2 THE IDEA OF THE PROOF OF THE REALIZATION THEOREM 3 SMALL DEFORMATIONS OF A POLYHEDRON 4 DEFORMATION OF A CONVEX POLYHEDRAL ANGLE 5 THE RIGIDITY THEOREM 6 REALIZABILITY OF THE METRICS CLOSE TO REALIZED METRICS 7 SMOOTH PASSAGE FROM A GIVEN METRIC TO A REALIZABLE METRIC 8 PROOF OF THE REALIZABILITY THEOREM CHAPTER VII EXISTENCE OF A CLOSED CONVEX SURFACE 1 THE RESULT AND THE METHOD OF PROOF 2 THE MAIN LEMMA ON CONVEX TRIANGLES 3 COROLLARIES OF THE MAIN LEMMA ON CONVEX TRIANGLES 4 THE COMPLETE ANGLE AT A POINT 5 CURVATURE AND TWO RELATED ESTIMATES 6 APPROXIMATION OF A METRIC OF POSITIVE CURVATURE 7 REALIZATION OF A METRIC OF POSITIVE CURVATURE GIVEN ON A SPHERE CHAPTER VIII OTHER EXISTENCE THEOREMS 1 THE GLUING THEOREM 2 APPLICATION OF THE GLUING THEOREM TO THE REALIZATION THEOREMS 3 REALIZABILITY OF A COMPLETE METRIC OF POSITIVE CURVATURE 4 MANIFOLDS ON WHICH A METRIC OF POSITIVE CURVATURE CAN BE GIVEN 5 THE QUESTION OF THE UNIQUENESS OF A CONVEX SURFACE WITH A GIVEN METRIC 6 VARIOUS DE?NITIONS OF A METRIC OF POSITIVE CURVATURE CHAPTER IX CURVES ON CONVEX SURFACES 1 THE DIRECTION OF A CURVE 2 THE SWERVE OF A CURVE 3 THE GENERAL GLUING THEOREM 4 CONVEX DOMAINS 5 QUASIGEODESICS 6 A CIRCLE CHAPTER X AREA 1 THE INTRINSIC DE?NITION OF AREA 2 THE EXTRINSIC–GEOMETRIC MEANING OF AREA 3 EXTREMAL PROPERTIES OF PYRAMIDS AND CONES CHAPTER XI THE ROLE OF SPECI?C CURVATURE 1 INTRINSIC GEOMETRY OF A SURFACE 2 INTRINSIC GEOMETRY OF A SURFACE OF BOUNDED SPECI?C CURVATURE 3 THE SHAPE OF A CONVEX SURFACE DEPENDING ON ITS CURVATURE CHAPTER XII GENERALIZATION 1 CONVEX SURFACES IN SPACES OF CONSTANT CURVATURE 2 REALIZATION THEOREMS IN SPACES OF CONSTANT CURVATURE 3 SURFACES OF INDE?NITE CURVATURE