The asymptotics of heat kernels on Riemannian foliations

We study the basic Laplacian of a Riemannian foliation on a compact manifold M by comparing it to the induced Laplacian on the basic manifold. We show that the basic heat kernel on functions has a particular asymptotic expansion along the diagonal of M x M as t --> 0, which is computable in terms of geometric invariants of the foliation. We generalize this result to include heat kernels corresponding to other transversally elliptic operators acting on the space of basic sections of a vector bundle, such as the basic Laplacian on k-forms and the square of a basic Dirac operator.