New PDF release: A Spinorial Approach to Riemannian and Conformal Geometry

By Jean-Pierre Bourguignon, Oussama Hijazi, Jean-louis Milhorat, Andrei Moroianu, Sergiu Moroianu

ISBN-10: 3037191368

ISBN-13: 9783037191361

The e-book offers an common and accomplished advent 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 elemental algebraic, geometrical, analytical and topological materials, a scientific research of the spectral houses of the Dirac operator on compact spin manifolds is performed. The classical estimates on eigenvalues and their restricting instances are mentioned subsequent, highlighting the delicate interaction of spinors and detailed geometric constructions. numerous purposes of those rules are offered, together with spinorial proofs of the confident Mass Theorem or the type of optimistic Kähler-Einstein touch manifolds. illustration idea is used to explicitly compute the Dirac spectrum of compact symmetric areas. The targeted positive aspects of the booklet contain a unified therapy of and conformal spin geometry (with specified emphasis at the conformal covariance of the Dirac operator), an summary with proofs of the idea of elliptic differential operators on compact manifolds in line with pseudodifferential calculus, a spinorial characterization of distinctive geometries, and a self-contained presentation of the representation-theoretical instruments wanted so one can understand spinors. This ebook may also help complex graduate scholars and researchers to get extra accustomed to this gorgeous, notwithstanding no longer sufficiently identified, area of arithmetic with nice relevance to either theoretical physics and geometry. A book of the eu Mathematical Society (EMS). disbursed in the Americas via the yankee Mathematical Society.

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Extra info for A Spinorial Approach to Riemannian and Conformal Geometry

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13. If ‰ D Œu; Q  and ˆ D Œu; Q ', define h‰; ˆi ´ . 37 to check that this does not depend on the choice of the spinorial frame u. 39 of Chapter 1, yields directly the following theorem. 6. If M is a spin manifold of real dimension n, the complex spinor bundle †M carries a real structure jR , for n  1; 2; 6; 7; 8 mod 8, and a quaternionic structure jQ , for n  2; 3; 4; 5; 6 mod 8. The Clifford product with real tangent vectors commutes with jR , for n  6; 7; 8 mod 8, and anti-commutes with jR , for n  1; 2 mod 8.

Ii) The conformal spin group CSpinn is the group Spinn  RC . 20. For n  2, the homomorphism f Spin  ´ Adj n is a nontrivial double covering of the special orthogonal group SOn . In particular for n  3, the group Spinn is the universal cover of SOn . Proof. 8) we know that the image of a nonzero vector by the map f Rn n f0g  Cln ! GLn AdW is the symmetry with respect to the hyperplane orthogonal to this vector. The image f Spin is then the group of even products of such symmetries, which by the of Adj n Cartan–Dieudonné theorem, is exactly the group SOn .

Recall  that the conformal group COC n is identified with SOn RC via the canonical isomorphism W SOn RC ! A; t / 7 ! 20 it is clear that, for n  3, the morphism W CSpinn ! a; t / 7 ! a/; is the universal cover of the conformal group. So we have the fundamental exact sequences  and f1g ! Z=2Z ! Spinn ! SOn ! f1g  f1g ! Z=2Z ! COC ! f1g: n Consider next the group homomorphism | W Spinn  U1 ! a; z/ 7 ! a/z; 26 1. Algebraic aspects where {W Cln ! Cln Š Cln ˝R C is the standard inclusion of Cln in Cln .

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A Spinorial Approach to Riemannian and Conformal Geometry by Jean-Pierre Bourguignon, Oussama Hijazi, Jean-louis Milhorat, Andrei Moroianu, Sergiu Moroianu

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