By Jean-Pierre Bourguignon, Oussama Hijazi, Jean-louis Milhorat, Andrei Moroianu, Sergiu Moroianu
The ebook provides an undemanding and accomplished creation to Spin Geometry, with specific emphasis at the Dirac operator, which performs a primary function in differential geometry and mathematical physics. After a self-contained presentation of the fundamental algebraic, geometrical, analytical and topological materials, a scientific learn of the spectral homes of the Dirac operator on compact spin manifolds is conducted. The classical estimates on eigenvalues and their proscribing situations are mentioned subsequent, highlighting the sophisticated interaction of spinors and distinctive geometric constructions. a number of functions of those principles are provided, together with spinorial proofs of the optimistic Mass Theorem or the type of confident Kähler-Einstein touch manifolds. illustration concept is used to explicitly compute the Dirac spectrum of compact symmetric areas. The precise positive factors of the ebook comprise a unified remedy of and conformal spin geometry (with specific emphasis at the conformal covariance of the Dirac operator), an outline with proofs of the idea of elliptic differential operators on compact manifolds according to pseudodifferential calculus, a spinorial characterization of unique geometries, and a self-contained presentation of the representation-theoretical instruments wanted with a view to recognize spinors. This e-book may also help complex graduate scholars and researchers to get extra accustomed to this gorgeous, although now not sufficiently identified, area of arithmetic with nice relevance to either theoretical physics and geometry. A book of the ecu Mathematical Society (EMS). allotted in the Americas by way of the yank Mathematical Society.
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Additional info for A Spinorial Approach to Riemannian and Conformal Geometry
Resp. spin structure/ on M . Proof. We will not treat here the real case, which is similar to the complex one, but just algebraically more involved. Let h; i be a Hermitian product on E with the property that the action of unit vector fields on E is unitary. 35, using a partition of unity. We consider first the case where the dimension n of M is even. PSOn M /x can be seen as an isomorphism uW Cln ! M /x ; 7 ! b/ 2 SOn U1 on SOn . For each x 2 M , we define Px D fvW †n ! a/ v. /; as well as the projection W Px !
31. The map Cln ˝†n ! /. /. / if n is odd, n is called Clifford multiplication of with . 32. resp. resp. Cl0n /. resp. Cln / are irreducible and equivalent. x/W †˙ n ! †n are 0 isomorphisms. Moreover, under the isomorphism Cln Š Cln 1 , the vector spaces †˙ n correspond to the two inequivalent irreducible representations of Cln 1 . 32 1. Algebraic aspects Proof. Note that the complex subalgebra generated by Spinn Cln is the even part Cl0n of Cln . Hence, two representations of Cl0n are irreducible or equivalent if and only if it is the case for their restrictions to Spinn .
By the SOn -invariance of the -operator, these eigenspaces are invariant under the action of SOn . 30 1. 2 The spinor representations. In this section we will show that by considering the standard complex representations of the Clifford algebras, the spin group Spinn inherits two irreducible complex representations if n is even and one if n is odd. As those representations do not descend to the group SOn , they are called spinor representations. These representations are of fundamental importance since, together with the standard irreducible representations of the group SOn , they constitute the “fundamental” (in the sense mentioned before) irreducible representations of the spin group.
A Spinorial Approach to Riemannian and Conformal Geometry by Jean-Pierre Bourguignon, Oussama Hijazi, Jean-louis Milhorat, Andrei Moroianu, Sergiu Moroianu