The wave trace of the basic Laplacian of a Riemannian foliation near a non-zero period

Given a compact boundaryless Riemannian manifold that admits a Riemannian foliation, recall that the space of leaf closures is a singular stratified space. Associated to this space is an operator called the basic Laplacian defined on the space of smooth functions that are constant on the leaves (and, hence, the closures of the leaves of the foliation). The corresponding basic spectrum is, under certain assumptions, an infinite subset of the spectrum of the ordinary laplacian. If the metric is bundle-like with respect to the foliation, the trace of the basic wave operator can be analyzed, and invariants of the basic spectrum can be computed. These invariants include the lengths of certain geodesic arcs which are orthogonal to the leaf closures, and from them, basic wave trace asymptotic expansions are derived. Using the connection between Riemannian foliations and manifolds being acted upon by a compact Lie group of isometries, $G$ , the wave trace for the $G$ -invariant spectrum of a $G$ -manifold is sketched out as a related result.     

The singularities of the wave trace of the basic Laplacian of a Riemannian foliation

We apply techniques of microlocal analysis to the study of the transverse geometry of Riemannian foliations in order to analyze spectral invariants of the basic Laplacian acting on functions on a Riemannian foliation with a bundle-like metric. In particular, we consider the trace of the basic wave operator when the mean curvature form is basic. We extend the concept of basic functions to distributions and demonstrate the existence of the basic wave kernel. The singularities of the trace of this basic wave kernel occur at the lengths of certain geodesic arcs which are orthogonal to the closures of the leaves of the foliation. In cases when the foliation has regular closure, a complete representation of the trace of the basic wave kernel can be computed for t≠0. Otherwise, a partial trace formula over a certain set of lengths of well-behaved geodesic arcs is obtained.     

The G-invariant spectrum and non-orbifold singularities

We consider the G-invariant spectrum of the Laplacian on an orbit space M/G where M is a compact Riemannian manifold and G acts by isometries. We generalize the Sunada–Pesce–Sutton technique to the G-invariant setting to produce pairs of isospectral non-isometric orbit spaces. One of these spaces is isometric to an orbifold with constant sectional curvature whereas the other admits non-orbifold singularities and therefore has unbounded sectional curvature. We conclude that constant sectional curvature and the presence of non-orbifold singularities are inaudible to the G-invariant spectrum.     

Geometry and Topology of Some Fibered Riemannian Manifolds

We investigate a principal G-bundle with G-invariant Riemannian metric on its total space. We derive formulas describing the Levi-Civita connection and curvatures in two-dimensional directions. We obtain estimates of the influence of properties of sectional curvatures to topological invariants of the bundle.     

A Note on the Infimum of Energy of Unit Vector Fields on a Compact Riemannian Manifold

Our main result in this paper establishes that if G is a compact Lie subgroup of the isometry group of a compact Riemannian manifold M acting with cohomogeneity one in M and either G has no singular orbits or the singular orbits of G have dimension at most n−3, then the unit vector field N orthogonal to the principal orbits of G is weakly smooth and is a critical point of the energy functional acting on the unit normal vector fields of M. A formula for the energy of N in terms of the of integral of the Ricci curvature of M and of the integral of the square of the mean curvature of the principal orbits of G is obtained as well. In the case that M is the sphere and G the orthogonal group it is known that that N is minimizer. It is an open question if N is a minimizer in general.     

Homogeneous Riemannian manifolds of positive Ricci curvature

We prove that a homogeneous effective spaceM=G/H, whereG is a connected Lie group andH⊂G is a compact subgroup, admits aG-invariant Riemannian metric of positive Ricci curvature if and only if the spaceM is compact and its fundamental group π₁(M) is finite (in this case any normal metric onG/H is suitable). This is equivalent to the following conditions: the groupG is compact and the largest semisimple subgroupLG⊂G is transitive onG/H. Furthermore, ifG is nonsemisimple, then there exists aG-invariant fibration ofM over an effective homogeneous space of a compact semisimple Lie group with the torus as the fiber. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 334–340, September, 1995.     

The G-Fredholm Property of the \bar{\partial} -Neumann Problem

Let G be a unimodular Lie group, X a compact manifold with boundary, and M be the total space of a principal bundle G→M→X so that M is also a strongly pseudoconvex complex manifold. In this work, we show that if G acts by holomorphic transformations in M, then the Laplacian on M has the following properties: The kernel of □ restricted to the forms Λ ^p,q with q>0 is a closed, G-invariant subspace in L ²(M,Λ ^p,q ) of finite G-dimension. Secondly, we show that if q>0, then the image of □ contains a closed, G-invariant subspace of finite G-codimension in L ²(M,Λ ^p,q ). These two properties taken together amount to saying that □ is a G-Fredholm operator. It is a corollary of the first property mentioned that the reduced L ²-Dolbeault cohomology spaces of M are finite G-dimensional for q>0. The boundary Laplacian □ _b has similar properties.      

Dynamical properties of groups of automorphisms on Heisenberg nilmanifolds

Dynamical properties of actions of groups of automorphisms on Heisenberg nilmanifolds H/Γ are studied. It is proved that such a group G has only finite or dense orbits if the induced action on the associated torus has the same property. This gives a partial answer to a question of Margulis. Moreover, the G-invariant (or even stationary) measures on H/Γ are determined.     

Best Constants in Sobolev Inequalities on Riemannian Manifolds in the Presence of Symmetries

Let (M,g) be a smooth compact Riemannian manifold, and G a subgroup of the isometry group of (M,g). We compute the value of the best constant in Sobolev inequalities when the functions are G-invariant. Applications to non-linear PDEs of critical or upper critical Sobolev exponent are also presented.     

Recognition of the Finite Simple Groups F 4(2m) by Spectrum

The spectrum of a finite group is the set of its element orders. A finite group G is said to be recognizable by spectrum, if every finite group with the same spectrum as G is isomorphic to G. The purpose of the paper is to prove that for every natural m the finite simple Chevalley group F ₄(2 ^m ) is recognizable by spectrum.     

The Ricci curvature of finite dimensional approximations to loop and path groups

Let W(G) and L(G) denote the path and loop groups respectively of a connected real unimodular Lie group G endowed with a left-invariant Riemannian metric. We study the Ricci curvature of certain finite dimensional approximations to these groups based on partitions of the interval [0,1]. We find that the Ricci curvatures of the finite dimensional approximations are bounded below independent of partition iff G is of compact type with an Ad-invariant metric.     

Equivariance and Multiplicity for the Scalar Curvature Problem

Given (M, g ₀) a three-dimensional compact Riemannian manifold, assumed not to be conformally diffeomorphic to the standard unit 3-sphere, and G a compactsubgroup of the conformal group of (M, g ₀), we first study conditions for a smooth G-invariant function f to be the scalar curvature of a G-invariant conformalmetric to g ₀. Then, extending previous results of Hebeyand Vaugon, we study conditions for f to be the scalarcurvature of at least two conformal metrics to g ₀.     

Geodesic symmetries of homogeneous Kähler Manifolds

D'Atri and Nickerson [6], [7] have given necessary conditions for the geodesic symmetries of a Riemannian manifold to preserve the volume element. We use their results to show that ifG is a compact simple Lie group,T is a maximal torus ofG, andG/T is not symmetric, then anyG-invariant Kähler metric onG/T does not have volume-preserving geodesic symmetries. From the Kähler/de Rham decomposition of a compact homogeneous Kähler manifold [8], our result extends to the invariant Kähler metrics on a quotient of a compact connected Lie group by a maximal torus. In proving these results we compute directly the Ricci tensor of anyG-invariant Kähler metric onG/T forG compact connected andT a maximal torus ofG. The result is an explicit formula giving the value of the Ricci tensor elements in terms of the root structure of the Lie algebra ofG.     

Surgery and equivariant Yamabe invariant

We consider the equivariant Yamabe problem, i.e., the Yamabe problem on the space of G-invariant metrics for a compact Lie group G. The G-Yamabe invariant is analogously defined as the supremum of the constant scalar curvatures of unit volume G-invariant metrics minimizing the total scalar curvature functional in their G-invariant conformal subclasses. We prove a formula about how the G-Yamabe invariant changes under the surgery of codimension 3 or more, and compute some G-Yamabe invariants.     

The signless Laplacian spectra of the corona and edge corona of two graphs

We give complete information about the signless Laplacian spectrum of the corona of a graph G ₁ and a regular graph G ₂, and complete information about the signless Laplacian spectrum of the edge corona of a connected regular graph G ₁ and a regular graph G ₂.     

Homogeneous spaces with polar isotropy

 We consider homogeneous spaces G/K with G a simple compact Lie group, endowed with an arbitrary G-invariant Riemannian metric. We classify those spaces where the action of K on G/K is polar and show that such spaces are locally symmetric. Moreover we give a classification of pairs (G,K) with G compact semisimple such that K has polar linear isotropy representation. Received: 16 May 2002 / Revised version: 8 November 2002 Published online: 3 March 2003 Mathematics Subject Classification (2000): 53C35, 57S15     

On the Discrete Spectrum of Generalized Quantum Tubes

The spectrum (of the Dirichlet Laplacian) of non-compact, non-complete Riemannian manifolds is much less understood than their compact counterparts. In particular it is often not even known whether such a manifold has any discrete spectra. In this article, we will prove that a certain type of non-compact, non-complete manifold called the quantum tube has non-empty discrete spectrum. The quantum tube is a tubular neighborhood built about an immersed complete manifold in Euclidean space. The terminology of “quantum” implies that the geometry of the underlying complete manifold can induce discrete spectra – hence quantization. We will show how the Weyl tube invariants appear in determining the existence of discrete spectra. This is an extension and generalization, on the geometric side, of the previous work of the author on the “quantum layer.”     

Construction and Properties of the t-Invariant

Special spine theory is used for constructing a new invariant of compact 3-manifolds: the t-invariant. The behavior of the invariant under (boundary) connected sum is investigated. One of the TuraevViro invariants is expressed via the t-invariant. The t-invariant is interpreted from the point of view of TQFT. The values of the t-invariant are computed for lens spaces and for all closed oriented 3-manifolds of complexity at most six. It is proved that the set of values of the t-invariant on Seifert manifolds with fixed base (which is a closed surface) and fixed number of singular fibers is finite. Bibliography: 10 titles.     

Kazhdan's property (T), L^2-spectrum and isoperimetric inequalities for locally symmetric spaces

Let V = G\G/KV =\Gamma\backslash G/K be a Riemannian locally symmetric space of nonpositive sectional curvature and such that the isometry group G of its universal covering space has Kazhdan's property (T). We establish strong dichotomies between the finite and infinite volume case. In particular, we characterize lattices (or, equivalently, arithmetic groups) among discrete subgroups G ì G\Gamma\subset G in various ways (e.g., in terms of critical exponents, the bottom of the spectrum of the Laplacian and the behaviour of the Brownian motion on V).     

An addition theorem for the manifolds with the Laplacian having discrete spectrum

The question of the preservation of discreteness of the spectrum of the Laplacian acting in a space of differential forms under the cutting and gluing of manifolds reduces to the same problem for compact solvability of the operator of exterior derivation. Along these lines, we give some conditions on a cut Y dividing a Riemannian manifold X into two parts X ₊ and X _? under which the spectrum of the Laplacian on X is discrete if and only if so are the spectra of the Laplacians on X ₊ and X _?.