# A Spinorial Approach to Riemannian and Conformal Geometry

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*Jean-Pierre Bourguignon; Oussama Hijazi; Jean-Louis Milhorat; Andrei Moroianu; Sergiu Moroianu*

A publication of the European Mathematical Society

The 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
\(\mathrm{Spin^c}\)
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). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in theoretical physics and geometry.