Applications of Clifford Modules
Clifford modules, i.e., representations of Clifford algebras, were already used in the structure theory of the Clifford algebra; see Chapter 8D. It is an amazing fact that Clifford algebras and Clifford modules over the real and complex numbers lie at the core of an astonishing variety of problems in differential geometry and topology. In particular, they play an important role in the analysis of vector fields on spheres, Lie groups and algebras, Bott periodicity, partial differential equations, immersions of manifolds into spheres, curvature properties in Riemannian geometry, and the structure of isoparametric submanifolds. This chapter will be nothing more than a sequence of glimpses at some of these connections. For details, refer to the impressive recent volumes by Berline-Getzler-Vergne, Gilbert-Murray, and Lawson-Michelsohn; to Kazdan’s survey article; and to Stolz  and Thorbergsson . The main references for the discussion that follows are Lawson-Michelsohn and Husemoller. In the analytical context the Clifford algebra is defined slightly differently, i.e., the requirement x2 =q(x) is replaced by x2 = -q(x). This leads to some notational differences between the references and the following discussion.
KeywordsDirac Operator Clifford Algebra Free Abelian Group Spin Manifold Positive Scalar Curvature
Unable to display preview. Download preview PDF.