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Dirac Operators in Riemannian Geometry
Thomas Friedrich, Institut für Mathematik, Humboldt-Universität, Berlin, Germany
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2000; 195 pp; hardcover
Volume: 25
ISBN-10: 0-8218-2055-9
ISBN-13: 978-0-8218-2055-1
List Price: US$42 Member Price: US$33.60
Order Code: GSM/25

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.

Graduate students and researchers in mathematics or physics.

Reviews

"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."

From a review of the German edition:

"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