The asymptotics of heat kernels on Riemannian foliations
Let \(M\) be an \(n\)-dimensional, connected, oriented Riemannian manifold. Assume that \(M\) is equipped with a Riemannian foliation \(\mathcal F\) of dimension \(q\). In the paper under review, the basic Laplacian of the foliation is compared with the induced Laplacian on the basic manifold. The author shows how to express an asymptotic expansion (of Minaksasundaram-Plejfel type) of the basic heat kernel on functions as \(t \rightarrow 0\), in terms of geometric invariants of the foliations. The result is generalized to the 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.
scientific article; zbMATH DE number 1156784