In this post, we will see the book Topics In Theory Of Surfaces In Elliptic Space by A. V. Pogorelov.

About the book

This book deals with the solution of a number of problems in the theory of surfaces in elliptic space,through consideration of isometric surfaces. The principal method of investigation is comparison of a pair of isometric figures in elliptic space with a pair of isometric figures in a Euclidean space which corresponds geodesically to the elliptic space. This enables us to transpose the main difficulties in the proof to Euclidean space, where they can be overcome by the appropriate theorems.

The book was translated from Russian by Rogey and Royer Inc. and was Richard Sacksteder edited by  published in 1961.

Credits to original uploader.

You can get the book here.

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Contents

Chapter I. Elliptic Space 1

1. Four-dimensional vector space 1
2. The concept of elliptic space 5
3. Curves in elliptic space 9
4. Surfaces in elliptic space 14
5. Fundamental equations in the theory of surfaces in elliptic space 18

Chapter II. Convex Bodies and Convex Surfaces in Elliptic Space 23

1. The concept of a convex body 23
2. Convex surfaces in elliptic space 35
3. The deviation of a segment on a convex surface from its semitangent at its initial point 30
4. Manifolds of curvature not less than K.A.D. Aleksandrov’s theorem 34

Chapter III. Transformation of Congruent Figures 41

1. Transformation of congruent figures in elliptic space to congruent figures in Euclidean space 41
2. Transformation of congruent figures in Euclidean space onto congruent figures in elliptic space 45
3. Transformation by infinitesimal motions 49
4. Transformation of straight lines and Planes 53

Chapter IV. Isometric Surfaces 59

1. Transformation of isometric surfaces 59
2. Transformation of locally convex isometric surfaces in elliptic space 63
3. Proof of lemma 1 67
4, Transformation of locally convex isometric surfaces in Euclidean space. 71
5. Proof of lemma 2 74

Chapter V. Infinitesimal Deformations of Surfaces in Elliptic Space 77

1. Pairs of isometric surfaces and infinitesimal deformations 77
2. Transformation of surfaces and their infinitesimal deformations 81
3. Some theorems on infinitesimal deformations of surfaces in elliptic space. 85

Chapter VI. Single-Value Definiteness of General Convex Surfaces in Elliptic Space 89

1. A lemma on rib points on a convex surface 89
2. Transformation of isometric dihedral angles and cones 92
3. Local convexity of the surfaces 𝜙_1 and 𝜙_2 at smooth points 97
4, Convexity and isometry of the surfaces 𝜙_1 and 𝜙_2 101
5. Various theorems on uniqueness of convex surfaces in elliptic space 105

Chapter VIL. Regularity of Convex Surfaces with a Regular Metric 113

1. The deformation equation for surfaces in elliptic space 113
2. Evaluation of the normal curvatures of a regular convex cap in elliptic space 117
3. Convex surfaces of bounded specific curvature in elliptic space 122
4. Proof of the regularity of convex surfaces with a regular metric in elliptic space 125

Bibliography 131