Are you between 5 and 25? If so, enter the Wicked Young Writer Awards NOW - click here for details...

debuts of the month
Search our site
A Country Escape by Katie Fforde Read the opening extract of the brand new Katie Fforde book before its publication on 22/02/2018

Dirac Operators in Riemannian Geometry by
  

Dirac Operators in Riemannian Geometry

Part of the Graduate Studies in Mathematics Series

Synopsis

Dirac Operators in Riemannian Geometry by

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 $\textrm {spin}^\mathbb{C}$) 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 $\textrm {spin}^\mathbb{C}$ 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.

About the Author

Loading other formats...

Book Info

Publication date

15th August 2000

Author



More books by
Author 'Like for Like'
    recommendations

Publisher

American Mathematical Society

Format

Hardback

Categories

Differential & Riemannian geometry
Topology
Calculus & mathematical analysis

ISBN

9780821820551

Lovereading always comes up with great suggestions and has introduced me to enjoyable books and new authors to discover.

Gaynor Passmore

Lovereading recommends, honestly reviews and promotes books-what more can I say?!

Rachel Bridgeman

I love reading books I wouldn't normally choose before everyone else gets to read them!

Dawn Lynch

Thanks to Lovereading I have discovered new writers and read books I would never had looked twice at - and enjoyed them.

Angela Rhodes

At Lovereading there are fabulous books available in every genre, with great reviews to help you pick the right book for you.

Teresa O'Halloran

I love the newsletter with reviews of all the new books coming out. Can't wait to open it when it arrives in my inbox.

Rachel Aygin

It's a lively, independent website with reviews, recommendations and more - with a huge range of books available to buy in all formats.

Alison Layland

Lovereading helps me decide what real people read.

Kerry Bridges

Lovereading4kids

Lovereading4schools