Bourguignon J.P., Hijazi O., Milhorat J.l., Moroianu A., Moroianu S. A Spinorial Approach to Riemannian and Conformal Geometry

European Mathematical Society, 2015. — 462 p. — (EMS Monographs in Mathematics). — ISBN10: 3037191368. — ISBN13: 978-3037191361The book gives an elementary and comprehensive introduction to Spin Geometry, with particular emphasis on the Dirac operator, which plays a fundamental role in differential geometry and mathematical physics. After a self-contained presentation of the basic algebraic, geometrical, analytical and topological ingredients, a systematic study of the spectral properties of the Dirac operator on compact spin manifolds is carried out. The classical estimates on eigenvalues and their limiting cases are discussed next, highlighting the subtle interplay of spinors and special geometric structures. Several applications of these ideas are presented, including spinorial proofs of the Positive Mass Theorem or the classification of positive Kähler-Einstein contact manifolds. Representation theory is used to explicitly compute the Dirac spectrum of compact symmetric spaces. The special features of the book include a unified treatment of and conformal spin geometry (with special emphasis on the conformal covariance of the Dirac operator), an overview with proofs of the theory of elliptic differential operators on compact manifolds based on pseudodifferential calculus, a spinorial characterization of special geometries, and a self-contained presentation of the representation-theoretical tools needed in order to apprehend spinors. This book will help advanced graduate students and researchers to get more familiar with this beautiful, though not sufficiently known, domain of mathematics with great relevance to both theoretical physics and geometry. A publication of the European Mathematical Society (EMS).Basic spinorial material
Algebraic aspects
Geometrical aspects
Topological aspects
Analytical aspects
Lowest eigenvalues of the Dirac operator on closed spin manifolds
Lower eigenvalue bounds on Riemannian closed spin manifolds
Lower eigenvalue bounds on Kähler manifolds
Lower eigenvalue bounds on quaternion-Kähler manifolds
Special spinor ELD and geometries
Special spinors on Riemannian manifolds
Special spinors on conformal manifolds
Special spinors on Kähler manifolds
Special spinors on quaternion-Kähler manifolds
Dirac spectra of model spaces
A brief survey on representation theory of compact groups
Symmetric space structure of model spaces
Riemannian geometry of model spaces
Explicit computations of the Dirac spectrum