An Introduction to Dirac Operators on Manifolds

A Dirac operator on a Riemannian manifold M acts on sections of a bundle S of left modules over the canonical Clifford bundle Cl(M). This construction allows for many special cases, a choice of the Dirac operator depends on a choice of the bundle S. The book is devoted to such cases, where the manifold M inherits its Riemannian structure from an embedding to a (pseudo)-Euclidean space Rp,q. The special case of embedded manifolds makes it possible to study various special constructions of the bundle S. The embedding of M into Rp,q also allows the use of the Clifford algebra Clp,q of Rp,q for various computations related to the normal bundle of M. The first chapter of the book treats Clifford algebras of a given vector space with a non-degenerate scalar product. Embedded manifolds, corresponding covariant derivatives and spinor fields are introduced in the second chapter. Various versions of Dirac operators are introduced and their properties are discussed in the third chapter. A description of conformal mappings using Clifford algebra 2 x 2 matrices and their relation to Dirac operators are given in Chapter 4. The next two chapters contain a discussion of the unique continuation property of solutions of a Dirac equation and their boundary values.

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