Ken Richardson

Publications and Preprints

The most recent items are listed first.

• (with Georges Habib) A brief note on the spectrum of the basic Dirac operator, to appear in the Bulletin of the London Mathematical Society. Click here to get a version from the arXiv.
  In this paper, we prove the invariance of the spectrum of the basic Dirac operator defined on a Riemannian foliation 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 ordinary Dirac operator. 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.
  
  
• (with Seoung Dal Jung) Transverse conformal Killing forms and a Gallot-Meyer Theorem for foliations, preprint. Click here to get a version from the arXiv.
  We study transverse conformal Killing forms on foliations and prove a Gallot-Meyer theorem for foliations. Moreover, we show that on a foliation with $C$-positive normal curvature, if there is a closed basic 1-form $\phi$ such that $\Delta_B\phi=qC\phi$, then the foliation is transversally isometric to the quotient of a $q$-sphere.
  
  
• (with Igor Prokhorenkov) Natural Equivariant Dirac Operators, to appear in the Journal of Geometry and Physics. Click here to get a version from the arXiv.
  We introduce a new class of natural, explicitly defined, transversally elliptic differential operators over manifolds with compact group actions. Under certain assumptions, the symbols of these operators generate all the possible values of the equivariant index. We also show that the components of the representation-valued equivariant index coincide with those of an elliptic operator constructed from the original data.
  
  
• Traces of heat kernels on Riemannian foliations, to appear in Transactions of the American Mathematical Society. This paper is available on the arXiv here.
  I consider the basic heat operator on forms of a Riemannian foliation on a compact manifold with a bundle--like metric, and I show that the trace T(t) of this operator has a particular asymptotic expansion as t approaches 0. The coefficients of the t^j and of t^j(log t)^k 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, I 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.
  Related work: Other researchers have studied the spectrum of operators in other singular situations. For example, in 1984 J. Br&uumlning and E. Heintze showed that the trace of the equivariant heat kernel (on a manifold with a compact group action by isometries) has a similar expansion. The problem of finding the trace of the basic heat kernel on a Riemannian foliation is reduced to finding the O(q)-invariant heat trace corresponding to a second-order, equivariant, elliptic differential operator on the basic manifold. The basic Weyl asymptotic formula and the existence of this heat kernel expansion may be used to show that the zeta function corresponding to the basic Laplacian may be meromorphically continued to the complex plane.
  
  
• (with Igor Prokhorenkov) Witten deformation and the equivariant index, Ann. Glob. Anal. Geom. 34(2008), no.3, pp. 301-327. (available online here ). Click here to get a version from the arXiv.
  Let $M$ be a compact Riemannian manifold endowed with an isometric action of a compact 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.
  
  
• Generalized equivariant index theory, Foliations 2005, 373--388, World Sci. Publ., Hackensack, NJ, 2006.
  
  
• (with Igor Prokhorenkov) Perturbations of Dirac operators, J. Geom. Phys. 57(2006), no. 1, pp. 297-321. Click here to get a version from the arXiv.
  We study general conditions under which the computations of the index of a perturbed Dirac operator $D_{s}=D+sZ$ localize to the singular set of the bundle endomorphism $Z$ in the semi-classical limit $s\to \infty $. 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. The paper contains a list of the current literature on the subject.
  
  
• (with Charles F. Bond, Jr) Seeing the Fisher Z-transformation, Psychometrika 69(2004), no. 2, pp. 291-303.
  
  
• (with Efton Park and Victor Belfi) A Hopf index theorem for foliations, Diff. Geom. Appl.18(2003), no. 3, 319-341. Click here to get a version from the arXiv.
  The Euler characteristic of a smooth, closed manifold may be defined as the alternating sum of the dimensions of the cohomology groups; we define the basic Euler characteristic of a smooth foliation to be the alternating sum of the dimensions of the basic cohomology groups. This definition makes sense for Riemannian foliations, since the dimensions of the cohomology groups are all finite. The classical Hopf index theorem states that, given a vector field with non-degenerate zeros on a smooth manifold, the Euler characteristic is the sum of the indices of that vector field. We prove an analogous result for the basic Euler characteristic and basic vector fields, but it is not possible to use the standard arguments using the Euler class or the Cech-deRham double complex. The problem is that many standard results such as the Poincare' lemma and Poincare' duality do not carry over to basic cohomology. However, with a few modifications, Witten's method of deforming the deRham complex does carry over to the foliation setting, and we are able to show that the basic Euler characteristic can be written as an alternating sum of basic Euler characteristics of the foliation restricted to leaf closures at which the basic field is tangent to the foliation. Thus, the basic Euler characteristic is an obstruction to the existence of a basic vector field that is never tangent to the (Riemannian) foliation.
  Related work: In 1992, A. Zeggar showed that for the case of taut Riemannian foliations, there exists a basic form that integrates to the basic Euler characteristic. We remark that no such form exists for Riemannian foliations that are not taut. In 1993, J. A. Alvarez-Lopez used Witten's method to prove a basic version of Morse theory, which is a special case of the theorem in this paper. We let the vector field be the gradient of the basic Morse function, and the result follows.
  
  
• (with Jeff Lee) Lichnerowicz and Obata theorems for foliations, Pacific J. Math. 206(2002), pp. 339-357.
  The standard Lichnerowicz comparison theorem (1958) states that, given an n-dimensional, closed Riemannian manifold with Ricci curvature satisfying Ric(X,X)\ge k(n-1)||X||² for every X in TM for some fixed k>0, then the smallest positive eigenvalue l of the Laplacian on M satisfies l \ge nk. The Obata theorem (1962) states that equality occurs if and only if M is isometric to the standard n-sphere of constant sectional curvature k. In this paper, we prove that if M is a Riemannian manifold with a Riemannian foliation of codimension q, and if the normal Ricci curvature (ie curvature is restricted to and contracted on the normal bundle of the foliation) satisfies Ricn(X,X) \ge k(q-1)||X||² for every X in the normal bundle for some fixed k>0, then the smallest positive eigenvalue l_B of the basic Laplacian on M satisfies l_B \ge qk. In addition, if equality occurs, then the leaf space of the foliation is isometric to the space of orbits of a discrete subgroup of O(q) acting on the standard q-sphere of constant curvature k. We also prove a result about bundle-like metrics: on any Riemannian foliation with bundle-like metric, there exists another bundle-like metric for which the mean curvature is basic and the basic Laplacian for the new metric is the same as that of the original metric.
  See comments on the related work to 4. above. Also, in 1986, Hebda showed that such a lower bound on the normal Ricci curvature implies that the Riemannian foliation is taut and that the transverse diameter of the foliation is bounded by \pi / \sqrt{k}.
  
  
• The transverse geometry of G-manifolds and Riemannian foliations, Illinois J. Math. 45(2001), pp. 517-535.
  I show that given a compact Riemannian manifold on which a compact Lie group acts by isometries, there exists a Riemannian foliation whose leaf closure space is naturally isometric (as a metric space) to the orbit space of the group action. Furthermore, this isometry (and foliation) may be chosen so that a leaf closure is mapped to an orbit with the same volume, even though the dimension of the orbit may be different from the dimension of the leaf closure. Conversely, given a Riemannian foliation, there is a metric on the basic manifold (an O(q)-manifold associated to the foliation) such that the leaf closure space is isometric to the O(q)-orbit space of the basic manifold via an isometry that preserves the volume of the leaf closures of maximal dimension. As a corollary, I show that every the orbit space of any Riemannian G-manifold is isometric to the orbit space of a Riemannian O(q)-manifold via an isometry that preserves the volumes of orbits of maximal dimension. Consequently, the spectrum of the basic Laplacian on functions on a Riemannian foliation (or the spectrum of the Laplacian on invariant functions on a G-manifold) may be identified with the spectrum of the Laplacian restricted to invariant functions on a Riemannian O(q)-manifold. Other similar results concerning the spectrum of differential operators on sections of vector bundles over Riemannian foliations and G-manifolds are discussed.
  Related work: In 1957 R. Palais showed that any such G-manifold may be imbedded in Euclidean space such that the group G acts by orthogonal transformations. It is well-known that every G-manifold may be locally modelled as an orthogonal group action. The results of this paper go one step further in showing that every G-manifold is transversally isometric to an O(q)-manifold. The theory of P. Molino (1986) gives a homeomorphism between the leaf closure space of a Riemannian foliation and the basic manifold; the results of this paper show that the metric on the basic manifold may be chosen so that the homeomorphism preserves the transverse geometry and transfers the basic analysis to invariant analysis.
  
  
• (with Jeff Lee) Riemannian foliations and eigenvalue comparison, Ann. Global Anal. Geom. 16(1998), 497-525.
  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. Cheng's eigenvalue comparison theorems assume bounds on the curvature of the manifold and then compare a specific eigenvalue to that 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 a foliation. The main theorems generalize Cheng's 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 first eigenvalue of such foliated tubes as well as some needed facts about the tubes themselves are established. This comparison theory, like Cheng's 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.
  Related work: Eigenvalue comparison theory on tubular domains in Riemannian manifolds has been studied by A. Gray, J. M. Lee, E. Heintze, H. Karcher, and others; the geometric analysis is more difficult than the case of metric balls. Eigenvalue comparison theorems have been proved by J. Barta, J. Cheeger, S. Y. Cheng, P. Li, H. P. McKean, M. Obata, S. T. Yau, and others. The results in this paper are the first known eigenvalue bounds for the basic Laplacian of a Riemannian foliation.
  
  
• The asymptotics of heat kernels on Riemannian foliations, Geom. Funct. Anal. 8(1998), 356-401.
  I study the basic Laplacian of a Riemannian foliation on a compact manifold M by comparing it to the induced Laplacian on the basic manifold associated to the foliation. I show that the basic heat kernel on functions has a particular asymptotic expansion along the diagonal of MxM, which is computable in terms of geometric invariants of the foliation. In contrast to the asymptotic expansion of the ordinary heat kernel of a Riemannian manifold, the nature of the expansion at x in M may depend on x, and the coefficients of the powers of t are not necessarily integrable. I generalize these results to include heat kernels corresponding to 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.
  Related work: The basic manifold is the leaf closure space of the lifted foliation on the orthonormal transverse frame bundle of the foliation; its existence and topological properties were shown by P. Molino in 1986, and its geometric structure is analyzed in this paper. The Minakshisundaram-Pleijel asymptotic expansion of the ordinary Laplacian of a Riemannian manifold was shown in 1949 and has been used extensively in global analysis since then. It is therefore natural to ask if the basic heat kernel of a Riemannian foliation has similar properties; this paper is the first to give a partial answer to this question.
  
  
• (with Efton Park) The basic Laplacian of a Riemannian foliation, Amer. J. Math. 118(1996), 1249-1275.
  The basic Laplacian of a foliation is the appropriate Laplacian that operates on basic forms - those forms that depend only on the transverse coordinates of the foliation. We write 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 and smoothness of the basic heat kernel and establish estimates for the eigenvalues of the basic Laplacian.
  Related work: B. Reinhart first defined Riemannian foliations in 1959, and J. A. Alvarez-Lopez, El Kacimi-Alaoui, G. Hector, F. Kamber, S. Nishikawa, M. Ramachandran, Ph. Tondeur, and many others have worked to analyze the basic Laplacian on such foliations. Kamber and Tondeur first proved a version of the basic Hodge theorem in 1986, and Nishikawa, Ramachandran, and Tondeur proved the existence of the basic heat kernel in 1990; however, these results required the assumption that the mean curvature vector field of the foliation is parallel along the leaves (or, in other words, the mean curvature form is a basic form). The results in our paper hold in complete generality.
  
  
• Critical points of the determinant of the Laplace operator, J. Funct. Anal. 122(1994), 52-83.
  This paper is based on the work in my Ph. D. thesis (Rice University, advisor: Robin Forman). Consider the zeta-function determinant of the Laplacian as a function on the set of metrics with fixed volume in the conformal class. In the work of Osgood, Philips, and Sarnak (1988), the researchers showed that for Riemann surfaces, the maximum of the determinant of the Laplacian occurs at the constant curvature metric. Thus, it is possible to use the determinant to uniformize the metric on surfaces. In this paper, I classify the critical points of the determinant of the ordinary Laplacian on higher dimensional manifolds. Surprisingly, the condition for a critical point is not a local condition (like constant scalar curvature) in any higher dimension, and I exhibit examples of manifolds of all possible higher dimensions with locally homogeneous metrics that are not critical points for the determinant functional. I also show that under certain conditions the determinant of the Laplacian on three-manifolds has a local maximum at these critical points; examples of local maxima include the standard metric on the three dimensional sphere and flat metrics on certain 3-tori.
  Related work: Many other researchers, such as Osgood, Phillips, and Sarnak, have studied the determinant. Some researchers avoid the non-local condition by studying instead the conformal Laplacian in higher dimensions, which has similar properties to the ordinary Laplacian on surfaces (see the work of S.-Y. Alice Chang, Paul Yang, Tom Branson, etc.). Kate Okikiolu later generalized my result, proving that the standard metric on the three-sphere is a local maximum of the determinant of the Laplacian with respect to smooth variations among all metrics of fixed volume.
  
  

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