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E-raamat: A.D. Alexandrov: Selected Works Part II: Intrinsic Geometry of Convex Surfaces

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A.D. Alexandrov: Selected Works Part II: Intrinsic Geometry of Convex SurfacesSuurem pilt
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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.

Arvustused

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

Foreword ix
Preface xi
Chapter I Basic Concepts and Results 1(56)
1. The General Concept and Problems of Intrinsic Geometry
1(7)
2. Gaussian Intrinsic Geometry
8(5)
3. A Polyhedral Metric
13(4)
4. Development
17(5)
5. Passage from Polyhedra to Arbitrary Surfaces
22(1)
6. A Manifold with an Intrinsic Metric
23(5)
7. Basic Concepts of Intrinsic Geometry
28(6)
8. Curvature
34(4)
9. Characteristic Properties of the Intrinsic Metric
38(8)
10. Some Singularities of the Intrinsic Geometry of Convex Surfaces
46(6)
11. Theorems of the Intrinsic Geometry of Convex Surfaces
52(5)
Chapter II General Propositions About Intrinsic Metrics 57(34)
1. General Theorems on Rectifiable Curves
57(7)
2. General Theorems on Shortest Arcs
64(7)
3. The Nonoverlapping Condition for Shortest Arcs
71(2)
4. A Convex Neighborhood
73(7)
5. General Properties of Convex Domains
80(3)
6. Triangulation
83(8)
Chapter III Characteristic Properties of the Intrinsic Metric 91(30)
1. Convergence of the Metrics of Convergent Convex Surfaces
91(8)
2. The Convexity Condition for a Polyhedral Metric
99(9)
3. The Convexity Condition for the Metric of a Convex Surface
108(5)
4. Consequences of the Convexity Condition
113(8)
Chapter IV The Angle 121(36)
1. General Theorems on Addition of Angles
121(7)
2. Theorems on Addition of Angles on Convex Surfaces
128(3)
3. The Angle of a Sector Bounded by Shortest Arcs
131(5)
4. On Convergence of Angles
136(5)
5. The Tangent Cone
141(6)
6. The Spatial Meaning of the Angle between Shortest Arcs
147(10)
Chapter V Curvature 157(40)
1. Intrinsic Curvature
157(6)
2. The Area of a Spherical Image
163(11)
3. Generalization of the Gauss Theorem
174(7)
4. The Curvature of a Borel Set
181(5)
5. Directions in Which It Is Impossible to Draw a Shortest Arc
186(2)
6. Curvature as Measure of Non-Euclidicity of the Metric
188(9)
Chapter VI Existence of a Convex Polyhedron with a Given Metric 197(38)
1. On Determining a Metric from a Development
197(7)
2. The Idea of the Proof of the Realization Theorem
204(6)
3. Small Deformations of a Polyhedron
210(3)
4. Deformation of a Convex Polyhedral Angle
213(5)
5. The Rigidity Theorem
218(4)
6. Realizability of the Metrics Close to Realized Metrics
222(3)
7. Smooth Passage from a Given Metric to a Realizable Metric
225(8)
8. Proof of the Realizability Theorem
233(2)
Chapter VII Existence of a Closed Convex Surface 235(42)
1. The Result and the Method of Proof
235(6)
2. The Main Lemma on Convex Triangles
241(8)
3. Corollaries of the Main Lemma on Convex Triangles
249(3)
4. The Complete Angle at a Point
252(6)
5. Curvature and Two Related Estimates
258(4)
6. Approximation of a Metric of Positive Curvature
262(7)
7. Realization of a Metric of Positive Curvature Given on a Sphere
269(8)
Chapter VIII Other Existence Theorems 277(24)
1. The Gluing Theorem
277(4)
2. Application of the Gluing Theorem to the Realization Theorems
281(3)
3. Realizability of a Complete Metric of Positive Curvature
284(4)
4. Manifolds on Which a Metric of Positive Curvature Can Be Given
288(7)
5. The Question of the Uniqueness of a Convex Surface with a Given Metric
295(3)
6. Various Definitions of a Metric of Positive Curvature
298(3)
Chapter IX Curves on Convex Surfaces 301(40)
1. The Direction of a Curve
301(7)
2. The Swerve of a Curve
308(8)
3. The General Gluing Theorem
316(4)
4. Convex Domains
320(6)
5. Quasigeodesics
326(6)
6. A Circle
332(9)
Chapter X Area 341(24)
1. The Intrinsic Definition of Area
341(9)
2. The Extrinsic-Geometric Meaning of Area
350(6)
3. Extremal Properties of Pyramids and Cones
356(9)
Chapter XI The Role of Specific Curvature 365(28)
1. Intrinsic Geometry of a Surface
365(11)
2. Intrinsic Geometry of a Surface of Bounded Specific Curvature
376(10)
3. The Shape of a Convex Surface Depending on Its Curvature
386(7)
Chapter XII Generalization 393(16)
1. Convex Surfaces in Spaces of Constant Curvature
393(5)
2. Realization Theorems in Spaces of Constant Curvature
398(4)
3. Surfaces of Indefinite Curvature
402(7)
Appendix Basics of Convex Bodies 409(16)
1. Convex Domains and Curves
409(2)
2. Convex Bodies. A Supporting Plane
411(3)
3. A Convex Cone
414(1)
4. Topological Types of Convex Bodies
415(3)
5. A Convex Polyhedron and the Convex Hull
418(3)
6. On Convergence of Convex Surfaces
421(4)
Index 425


S.S. Kutateladze
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