Spin geometry

Abstract: The book aims to give 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 Kahler-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 spin^c and conformal spin geometry (with special emphasis on the conformal covariance of the Dirac operator), an original introduction to pseudodifferential calculus, a spinorial characterization of special geometries, and a self-contained presentation of the representation-theoretical tools needed in order to apprehend spinors. We hope that 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.

Abstract: 1. Preliminaries 1.1. Introduction 1.2. What is a vector bundle? 1.3. What is a connection? 1.4. The curvature of a connection 1.5. Characteristic classes 1.6. The Thom form 1.7. The universal bundle 1.8. Classification of connections 1.9. Hodge theory 2. Spin geometry on four-manifolds 2.1. Euclidean geometry and the spin groups 2.2. What is a spin structure? 2.3. Almost complex and spin-c structures 2.4. Clifford algebras 2.5. The spin connection 2.6. The Dirac operator 2.7. The Atiyah-Singer index theorem 3. Global analysis 3.1. The Seiberg-Witten equations 3.2. The moduli space 3.3. Compactness of the moduli space 3.4. Transversality 3.5. The intersection form 3.6. Donaldson's theorem 3.7. Seiberg-Witten invariants 3.8. Dirac operators on Kaehler surfaces 3.9. Invariants of Kaehler surfaces Bibliography Index

Abstract: We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result directly extends to semi-Riemannian manifolds and to embeddings into spaces of constant curvature. We also give a new way to identify spinors for different metrics and to derive the variation formula for the Dirac operator. Moreover, we show that generalized Killing spinors for Codazzi tensors are restrictions of parallel spinors. Finally, we study the space of Lorentzian metrics and give a criterion when two Lorentzian metrics on a manifold can be joined in a natural manner by a 1-parameter family of such metrics.