Traces of heat operators on Riemannian foliations

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.Comment: 37 pages, citations update

Riemannian flows and adiabatic limits

We show the convergence properties of the eigenvalues of the Dirac operator on a spin manifold with a Riemannian flow when the metric is collapsed along the flow

A brief note on the spectrum of the basic Dirac operator

In this paper, we prove the invariance of the spectrum of the basic Dirac operator defined on a Riemannian foliation $(M,\mathcal{F})$ with respect to a change of bundle-like metric. We then establish new estimates for its eigenvalues on spin flows in terms of the O'Neill tensor and the first eigenvalue of the Dirac operator on $M$. We discuss examples and also define a new version of the basic Laplacian whose spectrum does not depend on the choice of bundle-like metric

The eta invariant on two-step nilmanifolds

The eta invariant appears regularly in index theorems but is known to be directly computable from the spectrum only in certain examples of locally symmetric spaces of compact type. In this work, we derive some general formulas useful for calculating the eta invariant on closed manifolds. Specifically, we study the eta invariant on nilmanifolds by decomposing the spin Dirac operator using Kirillov theory. In particular, for general Heisenberg three-manifolds, the spectrum of the Dirac operator and the eta invariant are computed in terms of the metric, lattice, and spin structure data. There are continuous families of geometrically, spectrally different Heisenberg three-manifolds whose Dirac operators have constant eta invariant. In the appendix, some needed results of L. Richardson and C. C. Moore are extended from spaces of functions to spaces of spinors.Comment: 53 pages, corrected final version, to appear in Communications in Analysis and Geometr