A.D. Alexandrov: Selected Works Part II: Intrinsic Geometry of Convex Surfaces (Classics of Soviet Mathematics) (Part 2)

A.D. Alexandrov's contribution to the field of intrinsic geometry was original and very influential. This text is a classic that remains unsurpassed in its clarity and scope. It presents his core material, originally published in Russian in 1948, beginning wth an outline of the main concepts and then exploring other topics, such as general propositions on an intrinsic metric; angles and curvature; existence of a convex polyhedron with prescribed metric; curves on convex surfaces; and the role of specific curvature. This text provides Adefinitive source for the development of intrinsic geometry and is indispensable for graduate students who want a better understanding of this subject. TABLE OF CONTENTS......Page 0 A.D. ALEXANDROV: SELECTED WORKS PART II: Intrinsic Geometry of Convex Surfaces......Page 2 CONTENTS......Page 4 FOREWORD......Page 7 PREFACE......Page 8 1. The General Concept and Problems of Intrinsic Geometry......Page 11 2. Gaussian Intrinsic Geometry......Page 18 3. A Polyhedral Metric......Page 23 4. Development......Page 27 5. Passage from Polyhedra to Arbitrary Surfaces......Page 32 6. A Manifold with an Intrinsic Metric......Page 33 7. Basic Concepts of Intrinsic Geometry......Page 38 8. Curvature......Page 44 9. Characteristic Properties of the Intrinsic Metric of a Convex Surface......Page 48 10. Some Singularities of the Intrinsic Geometry of Convex Surfaces......Page 56 11. Theorems of the Intrinsic Geometry of Convex Surfaces......Page 62 1. General Theorems on Rectifiable Curves......Page 67 2. General Theorems on Shortest Arcs......Page 74 3. The Nonoverlapping Condition for Shortest Arcs......Page 81 4. A Convex Neighborhood......Page 83 5. General Properties of Convex Domains......Page 90 6. Triangulation......Page 93 1. Convergence of the Metrics of Convergent Convex Surfaces......Page 101 2. The Convexity Condition for a Polyhedral Metric......Page 109 3. The Convexity Condition for the Metric of a Convex Surface......Page 118 4. Consequences of the Convexity Condition......Page 123 1. General Theorems on Addition of Angles......Page 131 2. Theorems on Addition of Angles on Convex Surfaces......Page 138 3. The Angle of a Sector Bounded by Shortest Arcs......Page 141 4. On Convergence of Angles......Page 146 5. The Tangent Cone......Page 151 6. The Spatial Meaning of the Angle between Shortest Arcs......Page 157 1. Intrinsic Curvature......Page 167 2. The Area of a Spherical Image......Page 173 3. Generalization of the Gauss Theorem......Page 184 4. The Curvature of a Borel Set......Page 191 5. The Set of Directions in Which It Is Impossible to Draw a Shortest Arc......Page 196 6. Curvature as a Measure of Non-Euclidicity of the Metric of a Surface......Page 198 1. On Determining a Metric from a Development......Page 206 2. The Idea of the Proof of the Realization Theorem......Page 213 3. Small Deformations of a Polyhedron......Page 219 4. Deformation of a Convex Polyhedral Angle......Page 222 5. The Rigidity Theorem......Page 227 6. Realizability of the Metrics Close to Realized Metrics......Page 231 7. Smooth Passage from a Given Metric to a Realizable Metric......Page 234 8. Proof of the Realizability Theorem......Page 242 1. The Result and the Method of Proof......Page 244 2. The Main Lemma on Convex Triangles......Page 250 3. Corollaries of the Main Lemma on Convex Triangles......Page 258 4. The Complete Angle at a Point......Page 261 5. Curvature and Two Related Estimates......Page 267 6. Approximation of a Metric of Positive Curvature......Page 271 7. Realization of a Metric of Positive Curvature Given on a Sphere......Page 278 1. The Gluing Theorem......Page 286 2. Application of the Gluing Theorem to the Realization Theorems......Page 290 3. Realizability of a Complete Metric of Positive Curvature......Page 293 4. Manifolds on Which a Metric of Positive Curvature Can Be Given......Page 297 5. The Question of the Uniqueness of a Convex Surface with a Given Metric......Page 304 6. Various Definitions of a Metric of Positive Curvature......Page 307 1. The Direction of a Curve......Page 310 2. The Swerve of a Curve......Page 317 3. The General Gluing Theorem......Page 325 4. Convex Domains......Page 329 5. Quasigeodesics......Page 335 6. A Circle......Page 341 1. The Intrinsic Definition of Area......Page 350 2. The Extrinsic–Geometric Meaning of Area......Page 359 3. Extremal Properties of Pyramids and Cones......Page 365 1. Intrinsic Geometry of a Surface......Page 373 2. Intrinsic Geometry of a Surface of Bounded Specific Curvature......Page 384 3. Shape of a Convex Surface Depending on Its Curvature......Page 394 1. Convex Surfaces in Spaces of Constant Curvature......Page 400 2. Realization Theorems in Spaces of Constant Curvature......Page 405 3. Surfaces of Indefinite Curvature......Page 409 1. Convex Domains and Curves......Page 415 2. Convex Bodies. A Supporting Plane......Page 417 3. A Convex Cone......Page 420 4. Topological Types of Convex Bodies......Page 421 5. A Convex Polyhedron and the Convex Hull......Page 424 6. On Convergence of Convex Surfaces......Page 427

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.

"A. D. 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."--BOOK JACKET "This classic is quite readable and opens a deeper understanding of this field also through self-study without any special prerequisites "--H. Rindler, Wien, in Monatshefte für Mathematik, Vol. 149, No. 4, 2006 Study of continuous length-preserving deformations of a surface is sensible not only for regular surfaces: it suffices that on a surface there be enough curves for which the concept of length is meaningful.