Riemannian Geometry

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Springer Science & Business Media, 2006/11/24 - 405 ページ

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

 

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目次

8 Exercises
183
The Bochner Technique
187
1 Killing Fields
188
2 Hodge Theory
202
3 Harmonic Forms
205
4 Clifford Multiplication on Forms
213
5 The Curvature Tensor
221
6 Further Study
229

1 Connections
22
2 The Connection in Local Coordinates
29
3 Curvature
32
4 The Fundamental Curvature Equations
41
5 The Equations of Riemannian Geometry
47
6 Some Tensor Concepts
51
7 Further Study
56
Examples
63
2 Warped Products
64
3 Hyperbolic Space
74
4 Metrics on Lie Groups
77
5 Riemannian Submersions
82
6 Further Study
90
Hypersurfaces
95
2 Existence of Hypersurfaces
97
3 The GaussBonnet Theorem
101
4 Further Study
107
5 Exercises
108
Geodesics and Distance
111
1 Mixed Partials
112
2 Geodesics
116
3 The Metric Structure of a Riemannian Manifold
121
4 First Variation of Energy
126
5 The Exponential Map
130
6 Why Short Geodesics Are Segments
132
7 Local Geometry in Constant Curvature
134
8 Completeness
137
9 Characterization of Segments
139
10 Riemannian Isometries
143
11 Further Study
149
Sectional Curvature Comparison I
153
2 Second Variation of Energy
158
3 Nonpositive Sectional Curvature
162
4 Positive Curvature
169
5 Basic Comparison Estimates
173
6 More on Positive Curvature
176
7 Further Study
182
Symmetric Spaces and Holonomy
235
1 Symmetric Spaces
236
2 Examples of Symmetric Spaces
244
3 Holonomy
252
4 Curvature and Holonomy
256
5 Further Study
262
6 Exercises
263
Ricci Curvature Comparison
265
2 Fundamental Groups and Ricci Curvature
273
3 Manifolds of Nonnegative Ricci Curvature
279
4 Further Study
290
Convergence
293
1 GromovHausdorff Convergence
294
2 Hölder Spaces and Schauder Estimates
301
3 Norms and Convergence of Manifolds
307
4 Geometric Applications
318
5 Harmonic Norms and Ricci curvature
321
6 Further Study
330
7 Exercises
331
Sectional Curvature Comparison II
333
2 Distance Comparison
338
3 Sphere Theorems
346
4 The Soul Theorem
349
5 Finiteness of Betti Numbers
357
6 Homotopy Finiteness
365
7 Further Study
372
De Rham Cohomology
375
2 Elementary Properties
379
3 Integration of Forms
380
4 Ćech Cohomology
383
5 De Rham Cohomology
384
6 Poincaré Duality
387
7 Degree Theory
389
8 Further Study
391
Bibliography
393
Index
397
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