**Graduate Studies in Mathematics**

Volume: 93;
2008;
494 pp;
Hardcover

MSC: Primary 53;

Print ISBN: 978-0-8218-2003-2

Product Code: GSM/93

List Price: $81.00

Individual Member Price: $64.80

**Electronic ISBN: 978-1-4704-1161-9
Product Code: GSM/93.E**

List Price: $81.00

Individual Member Price: $64.80

#### Supplemental Materials

# Topics in Differential Geometry

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*Peter W. Michor*

This book treats the fundamentals of differential geometry: manifolds,
flows, Lie groups and their actions, invariant theory, differential forms
and de Rham cohomology, bundles and connections, Riemann manifolds,
isometric actions, and symplectic and Poisson geometry.

The layout of the material stresses naturality and functoriality from the
beginning and is as coordinate-free as possible. Coordinate formulas are
always derived as extra information. Some attractive unusual aspects of this
book are as follows:

- Initial submanifolds and the Frobenius theorem for distributions of nonconstant rank (the Stefan-Sussman theory) are discussed.
- Lie groups and their actions are treated early on, including the slice theorem and invariant theory.
- De Rham cohomology includes that of compact Lie groups, leading to the study of (nonabelian) extensions of Lie algebras and Lie groups.
- The Frölicher-Nijenhuis bracket for tangent bundle valued differential forms is used to express any kind of curvature and second Bianchi identity, even for fiber bundles (without structure groups). Riemann geometry starts with a careful treatment of connections to geodesic structures to sprays to connectors and back to connections, going via the second and third tangent bundles. The Jacobi flow on the second tangent bundle is a new aspect coming from this point of view.
- Symplectic and Poisson geometry emphasizes group actions, momentum mappings, and reductions.

This book gives the careful reader working knowledge in a wide range of topics of modern coordinate-free differential geometry in not too many pages. A prerequisite for using this book is a good knowledge of undergraduate analysis and linear algebra.

#### Table of Contents

# Table of Contents

## Topics in Differential Geometry

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Preface ix10 free
- CHAPTER I. Manifolds and Vector Fields 114 free
- CHAPTER II. Lie Groups and Group Actions 4154
- CHAPTER III. Differential Forms and de Rham Cohomology 99112
- 8. Vector Bundles 99112
- 9. Differential Forms 113126
- 10. Integration on Manifolds 122135
- 11. De Rham Cohomology 129142
- 12. Cohomology with Compact Supports and Poincare Duality 139152
- 13. De Rham Cohomology of Compact Manifolds 151164
- 14. Lie Groups III. Analysis on Lie Groups 158171
- 15. Extensions of Lie Algebras and Lie Groups 169182

- CHAPTER IV. Bundles and Connections 191204
- CHAPTER V. Riemann Manifolds 273286
- CHAPTER VI. Isometric Group Actions or Riemann G-Manifolds 363376
- CHAPTER VII. Symplectic and Poisson Geometry 411424
- List of Symbols 477490
- Bibliography 479492
- Index 489502
- Back Cover Back Cover1510

#### Readership

Graduate students, research mathematicians and physicists interested in differential geometry, mechanics, and relativity.

#### Reviews

...remarkably effective. ... Michors book is a truly marvelous pick from which to learn a lot of beautiful, important, and current mathematics.

-- MAA Reviews

Throughout the book the author stresses the development of short exact sequences and takes evident delight in the applications that ensue. For the reviewer, this is one of the most enjoyable qualities of the text. The text is a treasure, and will open up to the diligent and patient reader a vast panorama of modern differential geometry.

Throughout the book the author stresses the development of short exact sequences and takes evident delight in the applications that ensue. For the reviewer, this is one of the most enjoyable qualities of the text. The text is a treasure, and will open up to the diligent and patient reader a vast panorama of modern differential geometry.

-- Mathematical Reviews