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The most recent
items are listed first.
- (with Jochen Brüning
and
Franz W. Kamber)
The eta invariant and equivariant index of transversally elliptic operators,
preprint.
Click here to
get a version from the
arXiv.
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.
- (with Lance Drager,
Jeff Lee,
and
Efton Park)
Smooth distributions are finitely generated,
Annals of Global Analysis and Geometry 41 (2012), no. 3, 357-369.
Click here for the online article.
Click here to
get a version from the
arXiv.
A subbundle
of variable dimension inside the tangent bundle of a
smooth manifold
is called a smooth distribution if it is
the pointwise span of a family of smooth vector fields.
We prove that all such distributions are finitely generated,
meaning that the family may be taken to be a finite collection.
Further, we show that the space of
smooth sections of such distributions need not be finitely
generated as a module over the smooth functions.
Our results are valid in greater generality, where the tangent
bundle may be replaced by an arbitrary vector bundle.
- (with Georges Habib)
Modified differentials and basic cohomology for Riemannian foliations,
to appear in Journal of Geometric Analysis. Online article available
here.
Click here to
get a version from the
arXiv.
We define a new version of the exterior derivative on the basic forms
of a Riemannian foliation to obtain a new form of basic cohomology
that satisfies Poincare' duality in the transversally orientable
case. We use this twisted basic cohomology to show relationships
between curvature, tautness, and vanishing of the basic Euler
characteristic and basic signature.
- (with Jochen Brüning
and
Franz W. Kamber)
Index theory for basic Dirac operators on Riemannian foliations,
Contemporary Mathematics 546 (2011), 39-81.
Click here to
get a version from the
arXiv.
In this paper we prove a formula for the analytic index of a basic Dirac-type
operator on a Riemannian foliation, solving a problem that has been open for
many years. We also consider more general indices given by twisting the basic
Dirac operator by a representation of the orthogonal group. The formula is a
sum of integrals over blowups of the strata of the foliation and also involves
eta invariants of associated elliptic operators. As a special case, a Gauss-Bonnet
formula for the basic Euler characteristic is obtained using two independent
proofs.
- (with Jochen Brüning
and
Franz W. Kamber)
The equivariant index theorem for transversally elliptic operators
and the basic index theorem for Riemannian foliations,
Electronic Research Announcements in Mathematical Sciences 17 (2010), 138-154.
Click here for the online link.
Click here to
get a version from the
arXiv.
In this expository paper, we explain 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 is an index formula for basic
Dirac operators on Riemannian foliations, a problem that was open for many years. This
paper summarizes the work in the papers arXiv:1005.3845 [math.DG] and
arXiv:1008.1757 [math.DG].
- (with Seoung Dal Jung and Keum Ran Lee)
Generalized Obata theorem and its applications on foliations,
J. Math. Anal. Appl. 376 (2011), 129-135.
Online article available here.
Click here to
get a version from the
arXiv.
We prove the generalized Obata theorem on foliations. Let M be a complete
Riemannian manifold with a foliation F of codimension q>1 and a bundle-like
metric. Then (M, F) is transversally isometric to the q-sphere of radius 1/c
in (q+1)-dimensional Euclidean space endowed with the action of a discrete
subgroup of the orthogonal group O(q), if and only if there exists a
non-constant basic function f such that \nabla_X df = -c^2 f X^\flat for all
basic normal vector fields X, where c is a positive constant and \nabla is the
connection on the normal bundle. By the generalized Obata theorem, we classify
such manifolds which admit transversal non-isometric conformal fields.
- (with Seoung Dal Jung)
Transverse conformal Killing forms and a Gallot-Meyer Theorem for foliations,
Mathematische Zeitschrift 270 (2012), no. 1-2, 337-350.
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, Geometriae Dedicata 151 (2011), no.1, 411-429.
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 operators on Riemannian foliations,
Trans. Amer. Math. Soc. 362 (2010), no. 5, 2301-2337. This paper is also
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 tj
and of tj(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. Brüning 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 Georges Habib)
A brief note on the spectrum of the basic Dirac operator, Bull.
London Math. Soc. 41(2009), 683-690.
Online article available here.
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 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
lB
of the basic Laplacian on M satisfies
lB \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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