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.