**Graduate Studies in Mathematics**

Volume: 25;
2000;
195 pp;
Hardcover

MSC: Primary 58;
Secondary 53; 57; 81

Print ISBN: 978-0-8218-2055-1

Product Code: GSM/25

List Price: $42.00

AMS Member Price: $33.60

MAA Member Price: $37.80

**Electronic ISBN: 978-1-4704-2080-2
Product Code: GSM/25.E**

List Price: $42.00

AMS Member Price: $33.60

MAA Member Price: $37.80

# Dirac Operators in Riemannian Geometry

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*Thomas Friedrich*

For a Riemannian manifold \(M\), the geometry, topology and analysis
are interrelated in ways that are widely explored in modern
mathematics. Bounds on the curvature can have significant
implications for the topology of the manifold. The eigenvalues of the
Laplacian are naturally linked to the geometry of the manifold. For
manifolds that admit spin (or \(\mathrm{spin}^\mathbb{C}\))
structures, one obtains further information from equations involving
Dirac operators and spinor fields. In the case of four-manifolds, for
example, one has the remarkable Seiberg-Witten invariants.

In this text, Friedrich examines the Dirac operator on Riemannian
manifolds, especially its connection with the underlying geometry and
topology of the manifold. The presentation includes a review of
Clifford algebras, spin groups and the spin representation, as well as
a review of spin structures and \(\mathrm{spin}^\mathbb{C}\) structures.
With this foundation established, the Dirac operator is defined and
studied, with special attention to the cases of Hermitian manifolds
and symmetric spaces. Then, certain analytic properties are
established, including self-adjointness and the Fredholm property.

An important link between the geometry and the analysis is provided
by estimates for the eigenvalues of the Dirac operator in terms of the
scalar curvature and the sectional curvature. Considerations of
Killing spinors and solutions of the twistor equation on \(M\) lead to
results about whether \(M\) is an Einstein manifold or conformally
equivalent to one. Finally, in an appendix, Friedrich gives a concise
introduction to the Seiberg-Witten invariants, which are a
powerful tool for the study of four-manifolds. There is also an
appendix reviewing principal bundles and connections.

This detailed book with elegant proofs is suitable as a text for
courses in advanced differential geometry and global analysis, and can
serve as an introduction for further study in these areas. This
edition is translated from the German edition published by Vieweg
Verlag.

#### Readership

Graduate students and researchers in mathematics or physics.

#### Reviews & Endorsements

This book is a nice introduction to the theory of spinors and Dirac operators on Riemannian manifolds … contains a nicely written description of the Seiberg-Witten theory of invariants for 4-dimensional manifolds … This book can be strongly recommended to anybody interested in the theory of Dirac and related operators.

-- European Mathematical Society Newsletter

This work is to a great extent a written version of lectures given by the author. As a consequence of this fact, the text contains full, detailed and elegant proofs throughout, all calculations are carefully performed, and considerations are well formulated and well motivated. This style is typical of the author. It is a pleasure to read the book; any beginning graduate student should have access to it.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Dirac Operators in Riemannian Geometry

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Introduction xi12 free
- Chapter 1. Clifford Algebras and Spin Representation 118 free
- 1.1. Linear algebra of quadratic forms 118
- 1.2. The Clifford algebra of a quadratic form 421
- 1.3. Clifford algebras of real negative definite quadratic forms 1027
- 1.4. The pin and the spin group 1431
- 1.5. The spin representation 2037
- 1.6. The group Spin[sup(C)] 2542
- 1.7. Real and quaternionic structures in the space of n-spinors 2946
- 1.8. References and exercises 3249

- Chapter 2. Spin Structures 3552
- Chapter 3. Dirac Operators 5774
- Chapter 4. Analytical Properties of Dirac Operators 91108
- Chapter 5. Eigenvalue Estimates for the Dirac Operator and Twistor Spinors 113130
- Appendix A. Seiberg-Witten Invariants 129146
- Appendix B. Principal Bundles and Connections 155172
- B.1. Principal fibre bundles 155172
- B.2. The classification of principal bundles 162179
- B.3. Connections in principal bundles 163180
- B.4. Absolute differential and curvature 166183
- B.5. Connections in U(1)-principal bundles and the Weyl theorem 169186
- B.6. Reductions of connections 173190
- B.7. Frobenius' theorem 174191
- B.8. The Freudenthal-Yamabe theorem 177194
- B.9. Holonomy theory 177194
- B.10. References 178195

- Bibliography 179196
- Index 193210
- Back Cover Back Cover1213