Affine Differential Geometry: Geometry of Affine Immersions

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This is a self-contained and systematic account of affine differential geometry from a contemporary viewpoint, not only covering the classical theory, but also introducing the modern developments that have happened over the last decade. In order both to cover as much as possible and to keep the text of a reasonable size, 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 last. Some of the important geometric surfaces considered are illustrated by computer graphics, making this a physically and mathematically attractive book for all researchers in differential geometry, and for mathematical physicists seeking a quick entry into the subject.
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Editorial Reviews

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"...a very beautiful book, which will be cherished by anyone who works in the field...very pleasant to read; the authors take their time to explain everything very clearly and always come to the point...I enjoyed reading this excellent book, and I can recommend it to every differential geometer." Franki Dillen, Mathematical Reviews
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Product Details

  • ISBN-13: 9780521064392
  • Publisher: Cambridge University Press
  • Publication date: 6/5/2008
  • Series: Cambridge Tracts in Mathematics Series, #111
  • Pages: 280
  • Product dimensions: 5.98 (w) x 8.98 (h) x 0.63 (d)

Table of Contents

Ch. I Affine geometry and affine connections
1 Plane curves 1
2 Affine space 7
3 Affine connections 11
4 Nondegenerate metrics 18
5 Vector bundles 22
Ch. II Geometry of affine immersions - the basic theory
1 Affine immersions 27
2 Fundamental equations. Examples 32
3 Blaschke immersions - the classical theory 40
4 Cubic forms 50
5 Conormal maps 57
6 Laplacian for the affine metric 64
7 Lelieuvre's formula 68
8 Fundamental theorem 73
9 Some more formulas 77
10 Laplacian of the Pick invariant 82
11 Behavior of the cubic form on surfaces 87
Ch. III Models with remarkable properties
1 Ruled affine spheres 91
2 Some more homogeneous surfaces 95
3 Classification of equiaffinely homogeneous surfaces 102
4 SL(n,R) and SL(n,R)/SO(n) 106
5 Affine spheres with affine metric of constant curvature 113
6 Cayley surfaces 119
7 Convexity, ovaloids, ellipsoids 122
8 Other characterizations of ellipsoids 125
9 Minkowski integral formulas and applications 129
10 The Blaschke-Schneider theorem 138
11 Affine minimal hypersurfaces and paraboloids 141
Ch. IV Affine-geometric structures
1 Hypersurfaces with parallel nullity 147
2 Affine immersions [actual symbol not reproducible] 152
3 The Cartan-Norden theorem 158
4 Affine locally symmetric hypersurfaces 161
5 Rigidity theorem of Cohn-Vossen type 165
6 Extensions of the Pick-Berwald theorem 169
7 Projective structures and projective immersions 174
8 Hypersurfaces in P[superscript n+1] and their invariants 181
9 Complex affine geometry 187
Notes 1. Affine immersions of general codimension 196
Notes 2. Surfaces in R[superscript 4] 198
Notes 3. Affine normal mappings 202
Notes 4. Affine Weierstrass formula 203
Notes 5. Affine Backlund transformations 209
Notes 6. Formula for a variation of ovaloid with fixed enclosed volume 213
Notes 7. Completeness and hyperbolic affine hyperspheres 215
Notes 8. Locally symmetric surfaces 217
Notes 9. Centro-affine immersions of codimension 2 221
Notes 10. Projective minimal surfaces in P[superscript 3] 230
Notes 11. Projectively homogeneous surfaces in P[superscript 3] 230
App. 1. Torsion, Ricci tensor, and projective invariants 235
App. 2. Metric, volume, divergence, Laplacian 240
App. 3. Change of immersions and transversal vector fields 242
App. 4. Blaschke immersions into a general ambient manifold 243
Bibliography 246
List of symbols 256
Index 258
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