Clifford Bundles and Clifford Algebras

In view of General Relativity, it is necessary to study physical fields, including solutions of the Dirac equation, in curved spacetimes. It is generally believed that the study of Riemannian (positive definite) metrics (infinitesimal distance functions) will ultimately be relevant to the more directly physical problem of Lorentz signature metrics, via principles of analytic continuation in signature. This adds impetus to the natural mathematical pursuit of studying spin structure, the Dirac operator, and other related operators on Riemannian manifolds. This lecture is a biased attempt at an introduction to this subject, with an emphasis on fundamental ideas likely to be important in future work, for example, Stein-Weiss gradients, Bochner-Weitzenböck formulas, the Hijazi inequality, and the Penrose local twistor idea. This provides at least a framework for the study of advanced topics such as spectral invariants and conformal anomalies, which are not treated here.