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

Posted by

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

The ebook supplies an user-friendly and finished creation to Spin Geometry, with specific emphasis at the Dirac operator, which performs a primary position 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 performed. The classical estimates on eigenvalues and their proscribing circumstances are mentioned subsequent, highlighting the delicate interaction of spinors and distinctive geometric buildings. numerous purposes of those principles are offered, together with spinorial proofs of the confident Mass Theorem or the type of confident Kähler-Einstein touch manifolds. illustration thought is used to explicitly compute the Dirac spectrum of compact symmetric areas. The distinct positive factors of the e-book comprise a unified remedy of and conformal spin geometry (with precise emphasis at the conformal covariance of the Dirac operator), an outline with proofs of the idea of elliptic differential operators on compact manifolds in keeping with pseudodifferential calculus, a spinorial characterization of certain geometries, and a self-contained presentation of the representation-theoretical instruments wanted with a view to understand spinors. This booklet might help complex graduate scholars and researchers to get extra conversant in this gorgeous, although no longer sufficiently identified, area of arithmetic with nice relevance to either theoretical physics and geometry. A ebook of the eu Mathematical Society (EMS). dispensed in the Americas by means of the yank Mathematical Society.

Show description

Read or Download A Spinorial Approach to Riemannian and Conformal Geometry PDF

Similar differential geometry books

Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsden

This booklet illustrates the vast variety of Jerry Marsden’s mathematical legacy in parts of geometry, mechanics, and dynamics, from very natural arithmetic to very utilized, yet continuously with a geometrical standpoint. every one contribution develops its fabric from the perspective of geometric mechanics starting on the very foundations, introducing readers to trendy concerns through illustrations in a variety of issues.

Geometry and Analysis on Manifolds: In Memory of Professor Shoshichi Kobayashi

This quantity is devoted to the reminiscence of Shoshichi Kobayashi, and gathers contributions from exotic researchers engaged on issues with reference to his examine components. The e-book is prepared into 3 elements, with the 1st half offering an summary of Professor Shoshichi Kobayashi’s occupation. this can be via expository direction lectures (the moment half) on fresh themes in extremal Kähler metrics and price distribution conception, with a view to be important for graduate scholars in arithmetic attracted to new themes in advanced geometry and intricate research.

Degenerate Complex Monge–Ampère Equations

Advanced Monge–Ampère equations were some of the most strong instruments in Kähler geometry due to the fact Aubin and Yau’s classical works, culminating in Yau’s option to the Calabi conjecture. A awesome program is the development of Kähler-Einstein metrics on a few compact Kähler manifolds. lately degenerate complicated Monge–Ampère equations were intensively studied, requiring extra complex instruments.

Extra resources for A Spinorial Approach to Riemannian and Conformal Geometry

Example text

Let us consider the standard complex representation  of the group SOn in the space Cn induced by the injective homomorphism SOn ,! SOn;C . 26. For any positive integer k such that k  m 1, if n D 2m, and k  m, if n D 2m C 1, the representation ƒk  in the space ƒk Cn is an irreducible representation of the group SOn . Proof. Let fe1 ; : : : ; en g be the canonical basis of Rn , identified with the canonical basis of Cn . The canonical basis of ƒk Cn is then given by the vectors eI D ei1 ^    ^ eik ; where I D fi1 <    < ik g runs through the set of k-element subsets of f1; : : : ; ng.

Q; q 0 / 7 ! x 7! qxq 0 W Sp1  Sp1 ! H/; 1 D qx qS0 /: It is easy to check that  is a group homomorphism with values in O4 and even SO4 , since Sp1  Sp1 is connected. 1; 1/g Š Z=2Z. Sp1  Sp1 / D SO4 , since the two groups have the same dimension. 6), the covering W SU2  SU2 7 ! A; B/ 7 ! X D y yN xN   2 H 7 ! AX Bx : t We now introduce the conformal spin group CSpinn ´ Spinn  RC . Recall  that the conformal group COC n is identified with SOn RC via the canonical isomorphism W SOn RC !

C /2 D 1; x  ! C D . 1/n 1 ! 11) We now prove the following two propositions. 29. 1 ˙ ! C /. Cl˙ n D   Cln D Cln  and  n/ D Cl . 2. Spin groups and their representations 31 Proof. Since .!  ˙ /2 D  ˙ ;    C D  C   D 0: ˙ Since n is odd, ! C and  ˙ are central in Cln . It is then clear that Cl˙ n D   Cln C are two ideals of Cln and Cln D Cln ˚ Cln . , ! Cl˙  n / D Cln and the two subalgebras are isomorphic. 30. For n odd, let n be a complex irreducible representation of Cln .

Download PDF sample

Rated 4.89 of 5 – based on 33 votes