Riemannian Geometry

1. Front Matter

   Pages i-xvi

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2. Riemannian Metrics

   • Peter Petersen

   Pages 1-18

3. Curvature

   • Peter Petersen

   Pages 19-61

4. Examples

   • Peter Petersen

   Pages 63-88

5. Hypersurfaces

   • Peter Petersen

   Pages 89-102

6. Geodesics and Distance

   • Peter Petersen

   Pages 103-136

7. Sectional Curvature Comparison I

   • Peter Petersen

   Pages 137-162

8. The Bochner Technique

   • Peter Petersen

   Pages 163-205

9. Symmetric Spaces and Holonomy

   • Peter Petersen

   Pages 207-242

10. Ricci Curvature Comparison

    • Peter Petersen

    Pages 243-271

11. Convergence

    • Peter Petersen

    Pages 273-315

12. Sectional Curvature Comparison II

    • Peter Petersen

    Pages 317-359

13. Back Matter

    Pages 361-434

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This book is meant to be an introduction to Riemannian geometry. The reader is assumed to have some knowledge of standard manifold theory, including basic theory of tensors, forms, and Lie groups. At times we shall also assume familiarity with algebraic topology and de Rham cohomology. Specifically, we recommend that the reader is familiar with texts like [14] or[76, vol. 1]. For the readers who have only learned something like the first two chapters of [65], we have an appendix which covers Stokes' theorem, Cech cohomology, and de Rham cohomology. The reader should also have a nodding acquaintance with ordinary differential equations. For this, a text like [59] is more than sufficient. Most of the material usually taught in basic Riemannian geometry, as well as several more advanced topics, is presented in this text. Many of the theorems from Chapters 7 to 11 appear for the first time in textbook form. This is particularly surprising as we have included essentially only the material students ofRiemannian geometry must know. The approach we have taken deviates in some ways from the standard path. First and foremost, we do not discuss variational calculus, which is usually the sine qua non of the subject. Instead, we have taken a more elementary approach that simply uses standard calculus together with some techniques from differential equations.

• Riemannian geometry
• Spinor
• Tensor
• curvature
• manifold

P. Petersen

Riemannian Geometry

"A nice introduction to Riemannian geometry, containing basic theory as well as several advanced topics."

—EUROPEAN MATHEMATICAL SOCIETY

• Department of Mathematics, University of California, Los Angeles, Los Angeles, USA

  Peter Petersen

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