An Introduction To Spin Geometry

Over the past two decades the geometry of spin manifolds and Dirac operators, and various associated index theorems have come to play an increasingly important role both in mathematics and in mathematical physics. In the area of differential geometry and topology they have become fundamental. Topics like spin cobordism, previously considered exotic even by topologists, are now known to play an essential role in such classical questions as the existence or non-existence of metrics of positive curvature. Indeed, the profound methods introduced into geometry by Atiyah, Bott, Singer and others are now indispensable to mathematicians working in the field. It is the intent of this paper to set out the fundamental concepts and to present these methods and results in unified way.A principal theme of the exposition here is the consistent use of Clifford algebra and their representations. This reflects the observed fact that these algebra emerge repeatedly at the very core of an astonishing variety of problems in geometry and topology.The object of this paper is to present the algebraic ideas which lie at the heart of spin geometry. The central concept is that of a Clifford algebra.Within the group of units of the algebra there is a distinguished subgroup,called the spin group.Our discussion begins in a very general algebraic context but soon moves to the real case in order to keep matters simple and in the domain of interest.