Ken Richardson

The basic Laplacian of a Riemannian foliation

We study the basic Laplacian on Riemannian foliations by writing the basic Laplacian in terms of the orthogonal projection from square-integrable forms to basic square-integrable forms. Using a geometric interpretation of this projection, we relate the ordinary Laplacian to the basic Laplacian. Among other results, we show the existence of the basic heat kernel and establish estimates for the eigenvalues of the basic Laplacian.

Seeing the FisherZ-transformation

Since 1915, statisticians have been applying FishersZ-transformation to Pearson product-moment correlation coefficients. We offer new geometric interpretations of this transformation.

Riemannian Foliations and Eigenvalue Comparison

Eigenvalue comparison theorems for the Laplacian on a Riemannian manifold generally give bounds for the first Dirichlet eigenvalue on balls in the manifold in terms of an eigenvalue arising from a geometrically or analytically simpler situation. Chengs eigenvalue comparison theory assumes bounds on the curvature of the manifold and then compares this eigenvalue to the eigenvalue of a ball in a constant curvature space form. In this paper we examine the basic Laplacian – the appropriate Laplacian on functions that are constant on the leaves of the foliation. The main theorems generalize Chengs eigenvalue comparison theorem and other eigenvalue comparison theorems to the category of Riemannian foliations by estimating the first Dirichlet eigenvalue for the basic Laplacian on a metric tubular neighborhood of a leaf closure. Several other facts about the the first eigenvalue of such foliated tubes as well as some needed facts about the tubes themselves are established. This comparison theory, like Chengs theorem, remains valid for large tubes that are not homotopic to the middle leaf closure and that may have irregular boundaries. We apply these results to obtain upper bounds for the eigenvalues of the basic Laplacian on a closed manifold in terms of curvature bounds and the transverse diameter of the foliation.

A Hopf index theorem for foliations

Abstract We formulate and prove an analog of the Hopf Index Theorem for Riemannian foliations. We compute the basic Euler characteristic of a closed Riemannian manifold as a sum of indices of a non-degenerate basic vector field at critical leaf closures. The primary tool used to establish this result is an adaptation to foliations of the Witten deformation method.

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 K B (t) of this operator has a particular asymptotic expansion as t → 0. The coefficients of t α and of t α < (log t) β 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 Weyls 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.

Perturbations of Dirac operators

Abstract We study general conditions under which the computations of the index of a perturbed Dirac operator D s = D + s Z localize to the singular set of the bundle endomorphism Z in the semiclassical limit s → ∞ . We show how to use Witten’s method to compute the index of D by doing a combinatorial computation involving local data at the nondegenerate singular points of the operator Z . In particular, we provide examples of novel deformations of the de Rham operator to establish new results relating the Euler characteristic of a spin c manifold to maps between its even and odd spinor bundles.

PERTURBATIONS OF BASIC DIRAC OPERATORS ON RIEMANNIAN FOLIATIONS

Using the method of Witten deformation, we express the basic index of a transversal Dirac operator over a Riemannian foliation as the sum of integers associated to the critical leaf closures of a given foliated bundle map.

WITTEN DEFORMATION AND THE EQUIVARIANT INDEX

Let M be a compact Riemannian manifold endowed with an isometric action of a compact, connected Lie group. The method of the Witten deformation is used to compute the virtual representation-valued equivariant index of a transversally elliptic, first order differential operator on M. The multiplicities of irreducible representations in the index are expressed in terms of local quantities associated to the isolated singular points of an equivariant bundle map that is locally Clifford multiplication by a Killing vector field near these points.

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.

Singular Riemannian flows and characteristic numbers

Let M be an even-dimensional, oriented closed manifold. We show that the restriction of a singular Riemannian flow on M to a small tubular neighborhood of each connected component of its singular stratum is foliated diffeomorphic to an isometric flow on the same neighborhood. We then prove a formula that computes characteristic numbers of M as the sum of residues associated with the infinitesimal foliation at the components of the singular stratum of the flow.