A.D. Alexandrov is considered by many to be the father of intrinsic geometry, second only to Gauss in surface theory. That appraisal stems primarily from this masterpiece--now available in its entirely for the first time since its 1948 publication in Russian. Alexandrov's treatise begins with an outline of the basic concepts, definitions, and results relevant to intrinsic geometry. It reviews the general theory, then presents the requisite general theorems on rectifiable curves and curves of minimum length. Proof of some of the general properties of the intrinsic metric of convex surfaces follows. The study then splits into two almost independent lines: further exploration of the intrinsic geometry of convex surfaces and proof of the existence of a surface with a given metric. The final chapter reviews the generalization of the whole theory to convex surfaces in the Lobachevskii space and in the spherical space, concluding with an outline of the theory of nonconvex surfaces. Alexandrov's work was both original and extremely influential. This book gave rise to studying surfaces "in the large," rejecting the limitations of smoothness, and reviving the style of Euclid. Progress in geometry in recent decades correlates with the resurrection of the synthetic methods of geometry and brings the ideas of Alexandrov once again into focus. This text is a classic that remains unsurpassed in its clarity and scope.
Author(s): S.S. Kutateladze
Edition: 1
Year: 2004
Language: English
Pages: 448
TABLE OF CONTENTS......Page 0
A.D. ALEXANDROV: SELECTED WORKS PART II: Intrinsic Geometry of Convex Surfaces......Page 3
CONTENTS......Page 5
FOREWORD......Page 8
PREFACE......Page 9
1. The General Concept and Problems of Intrinsic Geometry......Page 12
2. Gaussian Intrinsic Geometry......Page 19
3. A Polyhedral Metric......Page 24
4. Development......Page 28
5. Passage from Polyhedra to Arbitrary Surfaces......Page 33
6. A Manifold with an Intrinsic Metric......Page 34
7. Basic Concepts of Intrinsic Geometry......Page 39
8. Curvature......Page 45
9. Characteristic Properties of the Intrinsic Metric of a Convex Surface......Page 49
10. Some Singularities of the Intrinsic Geometry of Convex Surfaces......Page 57
11. Theorems of the Intrinsic Geometry of Convex Surfaces......Page 63
1. General Theorems on Rectifiable Curves......Page 68
2. General Theorems on Shortest Arcs......Page 75
3. The Nonoverlapping Condition for Shortest Arcs......Page 82
4. A Convex Neighborhood......Page 84
5. General Properties of Convex Domains......Page 91
6. Triangulation......Page 94
1. Convergence of the Metrics of Convergent Convex Surfaces......Page 102
2. The Convexity Condition for a Polyhedral Metric......Page 110
3. The Convexity Condition for the Metric of a Convex Surface......Page 119
4. Consequences of the Convexity Condition......Page 124
1. General Theorems on Addition of Angles......Page 132
2. Theorems on Addition of Angles on Convex Surfaces......Page 139
3. The Angle of a Sector Bounded by Shortest Arcs......Page 142
4. On Convergence of Angles......Page 147
5. The Tangent Cone......Page 152
6. The Spatial Meaning of the Angle between Shortest Arcs......Page 158
1. Intrinsic Curvature......Page 168
2. The Area of a Spherical Image......Page 174
3. Generalization of the Gauss Theorem......Page 185
4. The Curvature of a Borel Set......Page 192
5. The Set of Directions in Which It Is Impossible to Draw a Shortest Arc......Page 197
6. Curvature as a Measure of Non-Euclidicity of the Metric of a Surface......Page 199
1. On Determining a Metric from a Development......Page 207
2. The Idea of the Proof of the Realization Theorem......Page 214
3. Small Deformations of a Polyhedron......Page 220
4. Deformation of a Convex Polyhedral Angle......Page 223
5. The Rigidity Theorem......Page 228
6. Realizability of the Metrics Close to Realized Metrics......Page 232
7. Smooth Passage from a Given Metric to a Realizable Metric......Page 235
8. Proof of the Realizability Theorem......Page 243
1. The Result and the Method of Proof......Page 245
2. The Main Lemma on Convex Triangles......Page 251
3. Corollaries of the Main Lemma on Convex Triangles......Page 259
4. The Complete Angle at a Point......Page 262
5. Curvature and Two Related Estimates......Page 268
6. Approximation of a Metric of Positive Curvature......Page 272
7. Realization of a Metric of Positive Curvature Given on a Sphere......Page 279
1. The Gluing Theorem......Page 287
2. Application of the Gluing Theorem to the Realization Theorems......Page 291
3. Realizability of a Complete Metric of Positive Curvature......Page 294
4. Manifolds on Which a Metric of Positive Curvature Can Be Given......Page 298
5. The Question of the Uniqueness of a Convex Surface with a Given Metric......Page 305
6. Various Definitions of a Metric of Positive Curvature......Page 308
1. The Direction of a Curve......Page 311
2. The Swerve of a Curve......Page 318
3. The General Gluing Theorem......Page 326
4. Convex Domains......Page 330
5. Quasigeodesics......Page 336
6. A Circle......Page 342
1. The Intrinsic Definition of Area......Page 351
2. The Extrinsic–Geometric Meaning of Area......Page 360
3. Extremal Properties of Pyramids and Cones......Page 366
1. Intrinsic Geometry of a Surface......Page 374
2. Intrinsic Geometry of a Surface of Bounded Specific Curvature......Page 385
3. Shape of a Convex Surface Depending on Its Curvature......Page 395
1. Convex Surfaces in Spaces of Constant Curvature......Page 401
2. Realization Theorems in Spaces of Constant Curvature......Page 406
3. Surfaces of Indefinite Curvature......Page 410
1. Convex Domains and Curves......Page 416
2. Convex Bodies. A Supporting Plane......Page 418
3. A Convex Cone......Page 421
4. Topological Types of Convex Bodies......Page 422
5. A Convex Polyhedron and the Convex Hull......Page 425
6. On Convergence of Convex Surfaces......Page 428
