# Riemannian Geometry

Springer Science & Business Media, 24 de nov. de 2006 - 405 páginas

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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### Conteúdo

 Preface vii Riemannian Metrics 1 1 Riemannian Manifolds and Maps 2 2 Groups and Riemannian Manifolds 5 3 Local Representations of Metrics 8 4 Doubly Warped Products 13 5 Exercises 17 Curvature 21
 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 Direitos autorais

### Referências a este livro

 Introduction to Smooth ManifoldsVisualização parcial - 2003
 Riemannian Geometry: A Beginners Guide, Second EditionFrank MorganNão há visualização disponível - 1998
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