The purpose of this chapter is to show how one can transfer the results obtained earlier, particularly those in Chapters 3 and 4, from Euclidean space to a differentiable manifold.

As we have already seen in the Euclidean setting, it is important to distinguish between local (short time) and global (long time) aspects of the theory. When dealing with differentiable manifolds, this distinction becomes even more important. Indeed, by definition, all smooth manifolds are locally Euclidean, and so one should expect that, aside from a few technical details, there is no problem about transferring the local theory. When the manifold is compact, global theory is relatively easy. Namely, because it has nowhere to spread, the heat flow quickly equilibrates and so the fundamental solution to a non-degenerate (i.e., a is elliptic) Kolmogorov equation tends rapidly to its stationary state. On the other hand, when the manifold is non-compact, the global theory reflects the geometric growth properties of the particular manifold under consideration, and so the Euclidean case cannot be used to predict long time behavior.

For the reason just given, we will restrict our attention to the local theory. In fact, in order to avoid annoying questions about possible “explosion,” we will restrict our attention to compact manifolds, a restriction which, for the local theory, is convenient but inessential.