Traces of heat operators on Riemannian foliations

TL;DR

This paper studies the short-time asymptotics of the heat operator on functions over Riemannian foliations, linking the trace expansion coefficients to local geometric invariants and deriving spectral results.

Contribution

It establishes a short-time asymptotic expansion for the basic heat operator's trace on Riemannian foliations and connects coefficients to local transverse geometry, with explicit calculations for special cases.

Findings

Derived the asymptotic expansion of the heat trace.

Connected expansion coefficients to local transverse invariants.

Calculated first two coefficients for specific foliation types.

Abstract

We consider the basic heat operator on functions on a Riemannian foliation of a compact, Riemannian manifold, and we show that the trace of this operator has a particular short time asymptotic expansion. The coefficients in this expansion are obtainable from local transverse geometric invariants - functions computable by analyzing the manifold in an arbitrarily small neighborhood of a leaf closure. Using this expansion, we prove some results about the spectrum of the basic Laplacian, such as the analogue of Weyl's asymptotic formula. Also, we explicitly calculate the first two nontrivial coefficients of the expansion for special cases such as codimension two foliations and foliations with regular closure.