In this chapter, after a brief review of the basics of differential and Riemannian geometry, we shall discuss some of the fundamental concepts of noncommutative geometry. After that, we shall illustrate with examples how quantum dynamical semigroups arise naturally in the context of classical and noncommutative geometry, and how they carry important information about the underlying classical or noncommutative geometric spaces. These semigroups are essentially the ‘heat semigroups’ with unbounded generator given by the Laplacian (or some variant of it) on the underlying space; and in the classical case, the dilation of such semigroups naturally involve a suitable Brownian motion on the manifold. While the classical theory of heat semigroup and Brownian motion on a manifold is well established and quite rich, there is not yet any general theory of their counterparts in noncommutative geometry. Neither the theory of quantum stochastic calculus nor noncommutative geometry are at a stage for developing a general theory connecting the two. Instead of a general theory, the present state of both subjects calls for an understanding of various examples available, and this is what we try to do in this chapter. We do so at two levels: first, at the semigroup level, and then at the level of quantum stochastic processes coming from dilation of the semigroups.

Basics of differential and Riemannian geometry

We presume that the reader is familiar with the basic concepts of differential geometry, including tangent, cotangent, differential forms etc. and at least the definition and elementary properties of Lie groups. Let us very briefly review the concepts of connection, curvature and also of Riemannian geometry.