Affine differential geometry has undergone a period of revival and rapid progress in the past decade. This book is a self-contained and systematic account of affine differential geometry from a contemporary view. It covers not only the classical theory, but also introduces the modern developments of the past decade. The authors have concentrated on the significant features of the subject and their relationship and application to such areas as Riemannian, Euclidean, Lorentzian and projective differential geometry. In so doing, they also provide a modern introduction to the latter. Some of the important geometric surfaces considered are illustrated by computer graphics.
Author(s): Katsumi Nomizu, Takeshi Sasaki
Series: Cambridge Tracts in Mathematics
Publisher: Cambridge University Press
Year: 1995
Language: English
Pages: 276
Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Contents......Page 4
Preface......Page 6
Introduction......Page 8
1. Plane curves......Page 14
2. Affine space......Page 20
3. Affine connections......Page 24
4. Nondegenerate metrics......Page 31
5. Vector bundles......Page 35
1. Affine immersions......Page 40
2. Fundamental equations. Examples......Page 45
3. Blaschke immersions - the classical theory......Page 53
4. Cubic forms......Page 63
5. Conormal maps......Page 70
6. Laplacian for the affine metric......Page 77
7. Lelieuvre's formula......Page 81
8. Fundamental theorem......Page 86
9. Some more formulas......Page 90
10. Laplacian of the Pick invariant......Page 95
11. Behavior of the cubic form on surfaces......Page 100
1. Ruled affine spheres......Page 104
2. Some more homogeneous surfaces......Page 108
3. Classification of equiaffinely homogeneous surfaces......Page 115
4. SL(n,R) and SL(n, R)/SO(n)......Page 119
5. Affine spheres with affine metric of constant curvature......Page 126
6. Cayley surfaces......Page 132
7. Convexity, ovaloids, ellipsoids......Page 135
8. Other characterizations of ellipsoids......Page 138
9. Minkowski integral formulas and applications......Page 142
10. The Blaschke-Schneider theorem......Page 151
11. Affine minimal hypersurfaces and paraboloids......Page 154
1. Hypersurfaces with parallel nullity......Page 160
2. Affine immersions R^n \rightarrow R^{n+1}......Page 165
3. The Cartan-Norden theorem......Page 171
4. Affine locally symmetric hypersurfaces......Page 174
5. Rigidity theorem of Cohn-Vossen type......Page 178
6. Extensions of the Pick-Berwald theorem......Page 182
7. Projective structures and projective immersions......Page 187
8. Hypersurfaces in P^{n+1} and their invariants......Page 194
9. Complex affine geometry......Page 200
1. Affine immersions of general codimension......Page 209
2. Surfaces in R^4......Page 211
3. Affine normal mappings......Page 215
4. Affine Weierstrass formula......Page 216
5. Affine Backlund transformations......Page 222
6. Formula for a variation of ovaloid with fixed enclosed volume......Page 226
7. Completeness and hyperbolic affine hyperspheres......Page 228
8. Locally symmetric surfaces......Page 230
9. Centro-affine immersions of codimension 2......Page 234
11. Projectively homogeneous surfaces in p3......Page 243
1. Torsion, Ricci tensor, and projective invariants......Page 248
2. Metric, volume, divergence, Laplacian......Page 253
3. Change of immersions and transversal vector fields......Page 255
4. Blaschke immersions into a general ambient manifold......Page 256
Bibliography......Page 259
List of symbols......Page 269
Index......Page 271
