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 largely explored in smooth arithmetic. Bounds at the curvature may have major implications for the topology of the manifold. The eigenvalues of the Laplacian are certainly associated with the geometry of the manifold. For manifolds that admit spin (or $\textrm{spin}^\mathbb{C}$) constructions, one obtains extra details from equations concerning Dirac operators and spinor fields. in terms of four-manifolds, for instance, one has the amazing Seiberg-Witten invariants. during this textual content, Friedrich examines the Dirac operator on Riemannian manifolds, specifically its reference to the underlying geometry and topology of the manifold. The presentation contains a evaluate of Clifford algebras, spin teams and the spin illustration, in addition to a overview of spin constructions and $\textrm{spin}^\mathbb{C}$ buildings. With this starting place validated, the Dirac operator is outlined and studied, with unique recognition to the instances of Hermitian manifolds and symmetric areas. Then, convinced analytic homes are confirmed, together with self-adjointness and the Fredholm estate. a massive hyperlink among the geometry and the research is equipped by means of estimates for the eigenvalues of the Dirac operator when it comes to the scalar curvature and the sectional curvature. issues of Killing spinors and suggestions of the twistor equation on $M$ bring about effects approximately no matter if $M$ is an Einstein manifold or conformally resembling one. eventually, in an appendix, Friedrich offers a concise advent to the Seiberg-Witten invariants, that are a robust device for the examine of four-manifolds. there's additionally an appendix reviewing important bundles and connections. This designated booklet with dependent proofs is acceptable as a textual content for classes in complex differential geometry and international research, and will function an creation for extra research in those parts. This variation is translated from the German variation released by way of Vieweg Verlag.

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Then Q* is an SO(n)principal bundle over the space X*. We want to study the question of whether Q* admits a spin structure. Since 7r1 (P) = 7ri (X) = 1, for every group element y E r there exist two lifts -y4 of the transformation y : Q -* Q such that the following diagram commutes: P-+P Q7- Q If e is a left action of r on P with e(y) = then P* = r\P obviously becomes a spin structure on Q*. 2. Spin structures in covering spaces 43 on Q* and q : X --+ X* denotes the projection, then Q is the bundle q*(Q*) induced from Q* by means of this projection.

The equivalence classes of spin structures on an SO(n)-principal bundle Q over a connected CW-complex X are in bijective correspondence with those subgroups H C irl (Q) of index 2 which do not contain aF, aF 0 H. Proof. Let a subgroup H C 7r1(Q) of index 2 with aF 0 H be given. 1. Spin structures on SO(n)-principal bundles 37 with connected total space P. Fix a point po E P and denote by y Q x SO(n) -* Q the action of the group SO(n) on Q. The map P P x Spin(n) A AxA QxSO(n) µ- Q induces the homomorphism p# o (A x A)#, 71(P) = 7rl(P x Spin(n)) (n # 7ri(Q) ®7l(SO(n)) N-# i1(Q), the image in 7rl (Q) of which coincides with H.

0 0 ... -1 0 0 0 EZj _ j 1 0 ... 0 ... 0 0 Now we want to prove the formula '\*(ezej) = 2E,;j for X, : spin(n) = m2 --3 so(n). The path -y(t) = cos(t)+sin(t)eiej = -(cos(t/2)e;,+sin(t/2)ej)(cos(t/2)ei-sin(t/2)ei) is a subgroup y C Spin(n) with at (0) = eiej. Then, for k A('Y(t))ek = 616 i, j, 1. g. in the case k = i, we have A(y(t))ei = (cos(t) + sin(t)eiej)ei(cos(t) + sin(t)ejei) = cos2(t)ei + 2sin(t) cos(t)ej - sin2(t)ei = cos(2t)ei + sin(2t)ej. g(A(y(t))ei) = lei. This computation proves the formula A (eiei) _ Thus 2Eij .

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