Dirac Operators in Riemannian Geometry by Thomas Friedrich

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 evidently associated with the geometry of the manifold. For manifolds that admit spin (or $\textrm{spin}^\mathbb{C}$) buildings, one obtains additional details from equations concerning Dirac operators and spinor fields. in relation to four-manifolds, for instance, one has the amazing Seiberg-Witten invariants. during this textual content, Friedrich examines the Dirac operator on Riemannian manifolds, particularly its reference to the underlying geometry and topology of the manifold. The presentation features a evaluate of Clifford algebras, spin teams and the spin illustration, in addition to a overview of spin buildings and $\textrm{spin}^\mathbb{C}$ constructions. With this origin proven, the Dirac operator is outlined and studied, with particular consciousness to the circumstances of Hermitian manifolds and symmetric areas. Then, convinced analytic homes are tested, together with self-adjointness and the Fredholm estate. an incredible hyperlink among the geometry and the research is equipped by way of estimates for the eigenvalues of the Dirac operator by way of the scalar curvature and the sectional curvature. issues of Killing spinors and options of the twistor equation on $M$ bring about effects approximately even if $M$ is an Einstein manifold or conformally akin to one. ultimately, in an appendix, Friedrich offers a concise advent to the Seiberg-Witten invariants, that are a robust software for the examine of four-manifolds. there's additionally an appendix reviewing primary bundles and connections. This distinct booklet with based proofs is acceptable as a textual content for classes in complex differential geometry and worldwide research, and will function an creation for additional research in those parts. This variation is translated from the German variation released via Vieweg Verlag.

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8. References and exercises E. Artin. Geometric Algebra, Princeton University Press 1957. H. Baum. Spin-Strukturen and Dirac-Operatoren fiber pseudo-Riemannschen Mannigfaltigkeiten, Teubner Verlag Leipzig 1981. H. Baum, Th. Friedrich, R. Grunewald, I. Kath. Twistors and Killing Spinors on Riemannian Manifolds, Teubner Verlag 1991. 8. References and exercises 33 B. Budinich, A. Trautman. The Spinorial Chessboard, Springer-Verlag 1988. D. Husemoller. Fibre Bundles, McGraw-Hill 1966. B. -L. Michelsohn.

2. Spin Structures 42 Since 7rl (R) contains at most two elements, and n is even, in this case, 7r1(R) = 1. Summarizing, we obtain i (R) 7G2 1 if n is odd, if n is even. Since CIPn is simply connected, we can apply the corresponding criterion for the existence of a spin structure on R. The bundle R has a spin structure if and only if irl (R) = Z2. This leads to the Proposition. n admits a spin structure if and only if n is odd, n - 1 mod 2. Remark. It is important to note that two spin structures on one and the same SO(n)-principal bundle may not be equivalent even if the corresponding Spin(n)-principal bundles over X are equivalent.

1(F)\Spin(n + 1) as well as 7r1(A-1(r)\Spin(n+1)) _ A-' (r). Hence the sequence in question is 1 ) Z2 ) A-1(r) )r ) 1, and the manifold X* = r\Sn is a spin manifold if and only if this sequence splits. For a group F of odd order JFI this sequence always splits. Otherwise, the splitting of the above exact sequence is equivalent to the splitting of the sequence 1 , (A-1(F2) n Z2) ; A-1(F2) > F2 ) 1, where F2 C F is a 2-Sylow subgroup. Summarizing, we obtain the Proposition. Let F C SO(n + 1) be a finite subgroup acting freely on the sphere S.

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