By Thomas Friedrich

For a Riemannian manifold $M$, the geometry, topology and research are interrelated in ways in which are extensively explored in smooth arithmetic. Bounds at the curvature could have major implications for the topology of the manifold. The eigenvalues of the Laplacian are clearly associated with the geometry of the manifold. For manifolds that admit spin (or $\textrm{spin}^\mathbb{C}$) buildings, one obtains extra info from equations regarding Dirac operators and spinor fields. on the subject of four-manifolds, for instance, one has the impressive Seiberg-Witten invariants. during this textual content, Friedrich examines the Dirac operator on Riemannian manifolds, specially its reference to the underlying geometry and topology of the manifold. The presentation encompasses a assessment of Clifford algebras, spin teams and the spin illustration, in addition to a assessment of spin constructions and $\textrm{spin}^\mathbb{C}$ constructions. With this origin proven, the Dirac operator is outlined and studied, with exact cognizance to the circumstances of Hermitian manifolds and symmetric areas. Then, convinced analytic houses are demonstrated, together with self-adjointness and the Fredholm estate. a huge hyperlink among the geometry and the research is equipped via estimates for the eigenvalues of the Dirac operator when it comes to the scalar curvature and the sectional curvature. concerns of Killing spinors and ideas of the twistor equation on $M$ bring about effects approximately no matter if $M$ is an Einstein manifold or conformally comparable to one. ultimately, in an appendix, Friedrich provides a concise creation to the Seiberg-Witten invariants, that are a robust instrument for the research of four-manifolds. there's additionally an appendix reviewing valuable bundles and connections. This specific booklet with dependent proofs is acceptable as a textual content for classes in complicated differential geometry and international research, and will function an advent for extra examine in those parts. This version is translated from the German version released by way of Vieweg Verlag.

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