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

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

Homotopy invariance of cohomology and signature of a riemannian foliation

We prove that any smooth foliation that admits a Riemannian foliation structure has a well-defined basic signature, and this geometrically defined invariant is actually a foliated homotopy invariant. We also show that foliated homotopic maps between Riemannian foliations induce isomorphic maps on basic Lichnerowicz cohomology, and that the Alvarez class of a Riemannian foliation is invariant under foliated homotopy equivalence

Basic Dolbeault cohomology and Weitzenb\"ock frmulas on transversely K\"ahler foliations

We study basic Dolbeault cohomology and find new Weitzenb\"ock formulas on a transversely K\"ahler foliation. We investigate conditions on mean curvature and Ricci curvature that impose restrictions on basic Dolbeault cohomology. For example, we prove that on a transversely K\"ahler foliation with positive transversal Ricci curvature, there are no nonzero basic-harmonic forms of type $(r,0)$, among other results.Comment: 23 page

The eta invariant and equivariant index of transversally elliptic operators

We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a $G$-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications, we obtain an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years.Comment: 62 pages, typos correcte