Riemannian Geometry

Front Cover
Springer Science & Business Media, Nov 24, 2006 - Mathematics - 405 pages

Intended for a one year course, this volume serves as a single source, introducing students to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory, while also presenting the most up-to-date research. This book will appeal to readers with a knowledge of standard manifold theory, including such topics as tensors and Stokes theorem. Various exercises are scattered throughout the text, helping motivate readers to deepen their understanding of the subject.

Important additions to this new edition include:

* A completely new coordinate free formula that is easily remembered, and is, in fact, the Koszul formula in disguise;

* An increased number of coordinate calculations of connection and curvature;

* General fomulas for curvature on Lie Groups and submersions;

* Variational calculus has been integrated into the text, which allows for an early treatment of the Sphere theorem using a forgottten proof by Berger;

* Several recent results about manifolds with positive curvature.

From reviews of the first edition:

"The book can be highly recommended to all mathematicians who want to get a more profound idea about the most interesting

achievements in Riemannian geometry. It is one of the few comprehensive sources of this type."

- Bernd Wegner, Zentralblatt

 

Contents

Preface
1
Curvature
21
Examples
63
Hypersurfaces
93
Exercises
107
17
144
22
151
Sectional Curvature Comparison I
153
Distance Comparison
338
Sphere Theorems
346
82
349
Finiteness of Betti Numbers
357
Homotopy Finiteness
365
Further Study
372
Appendix De Rham Cohomology
375
Elementary Properties
379

29
177
8
183
The Bochner Technique
187
6
193
5
200
2
221
Symmetric Spaces and Holonomy
235
Ricci Curvature Comparison
265
63
275
74
290
Convergence
293
77
328
Sectional Curvature Comparison II
333
Integration of Forms
380
ˇCech Cohomology
383
De Rham Cohomology
384
90
386
Poincaré Duality
387
Degree Theory
389
Further Study
391
6
393
Index
397
331
399
107
402
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