Author(s): Su Buchin
Publisher: Science Press / Gordon and Breach
Year: 1983
Language: English
City: Beijing
Preface . i
Chapter I. Preliminaries . 1
§ 1. Transformation Groups and Subordinated Geometries . 1
§ 2. Affine Transformation Groups and Projective Transformation
Groups . 3
§ 3. Fundamental Theorems for Affine Plane Curves. 5
§ 4. Fundamental Theorems for Affine Space Curves. 12
§ 5. Outlines on the Theory of Surfaces in Affine Space. 17
Exercises and Theorems . 33
Chapter II. Some Global Problems in the Theory of Affine Plane
Curves. 36
§ 1. An Inequality of Blaschke . 36
§ 2. Minkowski-Bohmer Theorem . 41
§ 3. Sextactic Point Theorem . 43
§ 4. Two Theorems on Elliptically Curved Ovals . 47
§ 5. Isoperimetric Property for Ellipses . 56
§ 6. Three-Point Problem of Sylvester. 59
§ 7. The Greatest Property for Triangles . 61
Exercises and Theorems . 65
Chapter III. Geometrical Structure of Affine Surface Theory . 67
§ 1. Relation Between the Transon Planes and the Affine Surface-
Normal . 67
§ 2. Moutard Quadrics. 71
§ 3. Pairs of Asymptotic Osculating Quadrics. 82
§ 4. 6ech Transformations 2* and Their Applications . 87
Exercises and Theorems . 103
Chapter IV. Theory of Affine Moulding Surfaces and Affine Surfaces
of Revolution. 105
§ 1. Affine Moulding Surfaces and Their Transforms . 105
§ 2. Affine Surfaces of Revolution . 118
§ 3. Generalized Affine Moulding Surfaces and Affine Surfaces of
Revolution. 127
§ 4. Some Characterizations of Affine Surfaces of Revolution. 135
§ 5. New Treatment of Affine Surfaces of Revolution . 141
§ 6. An Extension of Affine Surfaces of Revolution . 145
Exercises and Theorems .
Chapter V. Some Relations Between Affine and Projective Surface
Theories . ^2
§ 1. Researches on the Classes of Surfaces Whose Canonical Lines
Are Affine Normals. 152
§ 2. Surfaces Iik) of the First Species . 158
§3. Surfaces 2'{*> of the Second Species. 162
§ 4. Representation of Asymptotically Isothermal Surfaces Z( ^. 168
§5. Surfaces . 1^
§ 6. Surfaces ,2(-1) . 19^
§ 7. Additional Notes to Surfaces 2(-1) . 209
Exercises and Theorems . 219
Appendices . 22 ^
1. Bonnet Problem in the Affine Theory of Surfaces. 220
2. Affine Moulding Hypersurfaces and Affine Hypersurfaces of Re¬
volution . 236
Books for Further Reading. 247
Index . 248
