Conformal differential geometry and its generalizations

✍ Scribed by Maks A. Akivis, Vladislav V. Goldberg

• Publisher: Wiley
• Year: 1996
• Tongue: English
• Leaves: 399
• Category: Library

✦ Synopsis

Comprehensive coverage of the foundations, applications, recent developments, and future of conformal differential geometry Conformal Differential Geometry and Its Generalizations is the first and only text that systematically presents the foundations and manifestations of conformal differential geometry. It offers the first unified presentation of the subject, which was established more than a century ago. The text is divided into seven chapters, each containing figures, formulas, and historical and bibliographical notes, while numerous examples elucidate the necessary theory. Clear, focused, and expertly synthesized, Conformal Differential Geometry and Its Generalizations * Develops the theory of hypersurfaces and submanifolds of any dimension of conformal and pseudoconformal spaces. * Investigates conformal and pseudoconformal structures on a manifold of arbitrary dimension, derives their structure equations, and explores their tensor of conformal curvature. * Analyzes the real theory of four-dimensional conformal structures of all possible signatures. * Considers the analytic and differential geometry of Grassmann and almost Grassmann structures. * Draws connections between almost Grassmann structures and web theory. Conformal differential geometry, a part of classical differential geometry, was founded at the turn of the century and gave rise to the study of conformal and almost Grassmann structures in later years. Until now, no book has offered a systematic presentation of the multidimensional conformal differential geometry and the conformal and almost Grassmann structures. After years of intense research at their respective universities and at the Soviet School of Differential Geometry, Maks A. Akivis and Vladislav V. Goldberg have written this well-conceived, expertly executed volume to fill a void in the literature. Dr. Akivis and Dr. Goldberg supply a deep foundation, applications, numerous examples, and recent developments in the field. Many of the findings that fill these pages are published here for the first time, and previously published results are reexamined in a unified context. The geometry and theory of conformal and pseudoconformal spaces of arbitrary dimension, as well as the theory of Grassmann and almost Grassmann structures, are discussed and analyzed in detail. The topics covered not only advance the subject itself, but pose important questions for future investigations. This exhaustive, groundbreaking text combines the classical results and recent developments and findings. This volume is intended for graduate students and researchers of differential geometry. It can be especially useful to those students and researchers who are interested in conformal and Grassmann differential geometry and their applications to theoretical physics.

✦ Table of Contents

Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Contents......Page 6
Introduction......Page 10
1.1 Conformal transformations and conformal spaces......Page 16
1.2 Moving frames in a conformal space......Page 23
1.3 Pseudoconformal spaces......Page 29
1.4 Examples of pseudoconformal spaces......Page 34
Notes......Page 43
2.1 Fundamental objects and tensors of a hypersurface......Page 46
2.2 Invariant normalization of hypersurfaces......Page 55
2.3 The rigidity theorem and the fundamental theorem......Page 60
2.4 Curvature lines of a hypersurface......Page 67
2.5 Geometric problems connected with the tensor c;j......Page 76
Notes......Page 85
3.1 Geometry of a submanifold in a conformal space......Page 88
3.2 Submanifolds carrying a net of curvature lines......Page 104
3.3 Submanifolds in a pseudoconformal space......Page 115
3.4 Line submanifolds of a three-dimensional projective space......Page 123
Notes......Page 130
4.1 A manifold with a conformal structure......Page 134
4.2 Weyl connections and Riemannian metrics compatible with a conformal structure......Page 147
4.3 A conformal structure on submanifolds of a conformal space......Page 156
4.4 A conformal structure on a hypersurface of a projective space......Page 165
Notes......Page 175
5.1 Structure equations of the CO(2, 2)-structure......Page 178
5.2 The CO(1, 3)-structure and the CO(4, 0)-structure......Page 184
5.3 The Hodge operator......Page 191
5.4 Completely isotropic submanifolds of four-dimensional conformal structures......Page 198
5.5 Four-dimensional webs and CO(2, 2)-structures......Page 208
5.6 Conformal structures of some metrics in general relativity......Page 217
5.7 Conformal structures on a four-dimensional hypersurface......Page 223
Notes......Page 232
CHAPTER 6 GEOMETRY OF THE GRASSMANN MANIFOLD......Page 236
6.1 Analytic geometry of the Grassmannian and the Grassmann mapping......Page 237
6.2 Geometry of the Grassmannian G(1, 4)......Page 247
6.3 Differential geometry of the Grassmannian......Page 251
6.4 Submanifolds of the Grassmannian G(m, n)......Page 259
6.5 Normalization of the Grassmann manifold......Page 267
6.6 Homogeneous normalization of the Grassmann manifold......Page 275
Notes......Page 280
7.1 Almost Grassmann structures on a differentiable manifold......Page 282
7.2 Structure equations and torsion tensor of an almost Grassmann manifold......Page 289
7.3 The complete structure object of an almost Grassmann manifold......Page 296
7.4 Manifolds endowed with semiintegrable almost Grassmann structures......Page 307
7.5 Multidimensional (p + 1)-webs and almost Grassmann structures associated with them......Page 316
7.6 Grassmann (p + 1)-webs......Page 320
7.7 Transversally geodesic and isoclinic (p + 1)-webs......Page 324
7.8 Grassmannizable d-webs......Page 329
Notes......Page 334
Bibliography......Page 338
Symbols Frequently Used......Page 370
Author Index......Page 374
Subject Index......Page 378
Back Cover......Page 399