Riemannian Foliations Research Articles

Taut submanifolds and foliations

We give an equivalent description of taut submanifolds of complete Riemannian manifolds as exactly those submanifolds whose normal exponential map has the property that every preimage of a point is a union of submanifolds. It turns out that every taut submanifold is also $\mathbb{Z}_2$-taut. We explicitly construct generalized Bott-Samelson cycles for the critical points of the energy functionals on the path spaces of a taut submanifold that, generically, represent a basis for the $\mathbb{Z}_2$-cohomology. We also consider singular Riemannian foliations all of whose leaves are taut. Using our characterization of taut submanifolds, we are able to show that tautness of a singular Riemannian foliation is actually a property of the quotient.

Tenseness of Riemannian flows

We show that any transversally complete Riemannian foliation eF&/em& of dimension one on any possibly non-compact manifold M is tense; namely, (M,eF&/em&) admits a Riemannian metric such that the mean curvature form of eF&/em& is basic. This is a partial generalization of a result of Dominguez, which says that any Riemannian foliation on any compact manifold is tense. Our proof is based on some results of Molino and Sergiescu, and it is simpler than the original proof by Dominguez. As an application, we generalize some well known results including Masa's characterization of tautness.

Localization of basic characteristic classes

We introduce basic characteristic classes and numbers as new invariants for Riemannian foliations. If the ambient Riemannian manifold M is complete, simply connected (or more generally if the foliation is a transversely orientable Killing foliation) and if the space of leaf closures is compact, then the basic characteristic numbers are determined by the infinitesimal dynamical behavior of the foliation at the union of its closed leaves. In fact, they can be computed with an Atiyah-Bott-Berline-Vergne-type localization theorem for equivariant basic cohomology.

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.

Higher Order Transverse Bundles and Riemannian Foliations

The purpose of this paper is to prove that each of the following conditions is equivalent to that the foliation ${\cal F}$ is riemannian: 1) the lifted foliation ${\cal F}^{r}$ on the $r$-transverse bundle $\nu ^{r}{\cal F}$ is riemannian for an $r\geq 1$; 2) the foliation ${\cal F}_{0}^{r}$ on a slashed $\nu_{\ast}^{r}{\cal F}$ is riemannian and vertically exact for an $r\geq 1$; 3) there is a positively admissible transverse lagrangian on a $\nu_{\ast}^{r}{\cal F}$, for an $r\geq 1$. Analogous results have been proved previously for normal jet vector bundles.

Progress in the theory of singular Riemannian foliations

A singular foliation is called a singular Riemannian foliation (SRF) if every geodesic that is perpendicular to one leaf is perpendicular to every leaf it meets. A typical example is the partition of a complete Riemannian manifold into orbits of an isometric action.In this survey, we provide an introduction to the theory of SRFs, leading from the foundations to recent developments. Sketches of proofs are included and useful techniques are emphasized. We study the local structure of SRFs in general and under curvature conditions in particular. We also review the solution of the Palais–Terng problem on integrability of the horizontal distribution. Important special classes of SRFs, like polar and variationally complete foliations and their relations, are treated. A characterization of SRFs whose leaf space is an orbifold is given. Moreover, desingularizations of SRFs are studied and applications, e.g., to Molinoʼs conjecture, are presented.

Théorèmes de type Obata pour certains feuilletages riemanniens

A theorem of Obata states that a complete Riemannian manifold of dimension n⩾2 admitting a nontrivial function f satisfying the differential equation Agradf=fIn is a sphere (Obata, 1962). In this paper, we propose to study the analogous situation for some Riemannian foliations of dimension 1 or codimension 1 on Riemannian manifolds of dimension n⩾3.

Geometry of orbit spaces of proper Lie groupoids

Abstract In this paper, we study geometric properties of quotient spaces of proper Lie groupoids. First, we construct a natural stratification on such spaces using an extension of the slice theorem for proper Lie groupoids of Weinstein and Zung. Next, we show the existence of an appropriate metric on the groupoid which gives the associated Lie algebroid the structure of a singular riemannian foliation. With this metric, the orbit space inherits a natural length space structure whose properties are studied. Moreover, we show that the orbit space of a proper Lie groupoid can be triangulated. Finally, we prove a de Rham theorem for the complex of basic differential forms on a proper Lie groupoid.

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 index of a transverse Dirac-type operator: the case of abelian Molino sheaf

We give a local formula for the index of a transverse Dirac-type operator on a compact manifold with a Riemannian foliation, under the assumption that the Molino sheaf is a sheaf of abelian Lie algebras.

The automorphism groups of foliations with transverse linear connection

AbstractThe category of foliations is considered. In this category morphisms are differentiable maps sending leaves of one foliation into leaves of the other foliation. We prove that the automorphism group of a foliation with transverse linear connection is an infinite-dimensional Lie group modeled on LF-spaces. This result extends the corresponding result of Macias-Virgós and Sanmartín Carbón for Riemannian foliations. In particular, our result is valid for Lorentzian and pseudo-Riemannian foliations.

Serre fibrations in the Morita category of topological groupoids

In this paper, we generalize the notion of Serre fibration to the Morita category of topological groupoids and derive the associated long exact sequence of homotopy groups. We use this results for calculation of homotopy groups of various groupoids, such as the foliation groupoid of a Riemannian foliation.

Integer points in domains and adiabatic limits

We prove an asymptotic formula for the number of integer points in a family of bounded domains in the Euclidean space with smooth boundary, which remain unchanged along some linear subspace and stretch out in the directions, orthogonal to this subspace. A more precise estimate for the remainder is obtained in the case when the domains are strictly convex. Using these results, we improved the remainder estimate in the adiabatic limit formula (due to the first author) for the eigenvalue distribution function of the Laplace operator associated with a bundle-like metric on a compact manifold equipped with a Riemannian foliation in a particular case when the foliation is a linear foliation on the torus and the metric is the standard Euclidean metric on the torus.

A conformal integral invariant on Riemannian foliations

Let M M be a closed manifold which admits a foliation structure F \mathcal {F} of codimension q ≥ 2 q\geq 2 and a bundle-like metric g 0 g_0 . Let [ g 0 ] B [g_0]_B be the space of bundle-like metrics which differ from g 0 g_0 only along the horizontal directions by a multiple of a positive basic function. Assume Y Y is a transverse conformal vector field and the mean curvature of the leaves of ( M , F , g 0 ) (M,\mathcal {F},g_0) vanishes. We show that the integral ∫ M Y ( R g T T ) d μ g \int _MY(R^T_{g^T})d\mu _g is independent of the choice of g ∈ [ g 0 ] B g\in [g_0]_B , where g T g^T is the transverse metric induced by g g and R T R^T is the transverse scalar curvature. Moreover if q ≥ 3 q\geq 3 , we have ∫ M Y ( R g T T ) d μ g = 0 \int _MY(R^T_{g^T})d\mu _g=0 for any g ∈ [ g 0 ] B g\in [g_0]_B . However there exist codimension 2 2 minimal Riemannian foliations ( M , F , g ) (M,\mathcal {F},g) and transverse conformal vector fields Y Y such that ∫ M Y ( R g T T ) d μ g ≠ 0 \int _MY(R^T_{g^T})d\mu _g\neq 0 . Therefore, ∫ M Y ( R g T T ) d μ g \int _MY(R^T_{g^T})d\mu _g is a nontrivial obstruction for the transverse Yamabe problem on minimal Riemannian foliation of codimension 2 2 .

Riemannian foliations and the kernel of the basic Dirac operator

Abstract In this paper, in the special setting of a Riemannian foliation en- dowed with a bundle-like metric, we obtain conditions that force the vanishing of the kernel of the basic Dirac operator associated to the metric; this way we extend the traditional setting of Riemannian foli- ations with basic-harmonic mean curvature, where Bochner technique and vanishing results are known to work. Beside classical conditions concerning the positivity of some curvature terms we obtain new rela- tions between the mean curvature form and the kernel of the basic Dirac operator

A vanishing result for the Spin c Dirac operator defined along the leaves of a foliation

In the setting of a closed Riemannian manifold endowed with a smooth, non-necessarily integrable distribution, we extend a Lichnerowicz type formula which is known to work in the particular case of a transverse bundle associated to a Riemannian foliation. Interesting settings in which non-integrable distributions appear naturally are emphasized. As an application, we consider the distribution as being even dimensional and integrable; we consider also a hermitian line bundle, with a hermitian connection, such that the induced curvature tensor is non-degenerate, and an arbitrary hermitian bundle endowed also with a hermitian connection. Taking the k power of the line bundle and canonically constructing a Spinc Dirac operator defined along the leaves of the foliation generated by the distribution, we prove a vanishing result for the half kernel of this operator.

On the Dirac spectrum of Riemannian foliations admitting a basic parallel 1-form

In this paper, in the special setting of a Riemannian foliation with basic, non-necessarily harmonic mean curvature we introduce a Weitzenböck–Lichnerowicz type formula which allows us to apply the classical Bochner–Lichnerowicz technique. We show that the lower bound for the eigenvalues of the basic Dirac operator can be calculated using only classical techniques. As another application, for general Riemannian foliations we calculate the above eigenvalue bound in the presence of a basic parallel 1-form, as an extension of a known result on a closed Riemannian manifold. Some results concerning the limiting case are obtained in the final part of the paper.

Continuity of the Álvarez class under deformations

A manifold M with a foliation ℱ is minimizable if there exists a Riemannian metric g on M such that every leaf of ℱ is a minimal submanifold of ( M , g ). For a closed manifold M with a Riemannian foliation ℱ, Álvarez López [1] defined a cohomology class of degree 1 called the Álvarez class whose triviality characterizes the minimizability of ( M , ℱ). In this paper, we show that the family of the Álvarez classes of a smooth family of Riemannian foliations on a closed manifold is continuous with respect to the parameter. The Álvarez class has algebraic rigidity under certain topological conditions on ( M , ℱ) as the author showed in [21]. As a corollary of these two results, we show that under the same topological conditions the minimizability of Riemannian foliations is invariant under deformations.

Modified Differentials and Basic Cohomology for Riemannian Foliations

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 Poincaré 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.

A REMARK ON THE BRYLINSKI CONJECTURE FOR ORBIFOLDS

AbstractThe paper presents a proof of the Brylinski conjecture for compact Kähler orbifolds. The result is a corollary of the foliated version of the Mathieu theorem on symplectic harmonic representations of de Rham cohomology classes. The proofs are based on the idea of representing an orbifold as the leaf space of a Riemannian foliation and on the correspondence between foliated and holonomy invariant objects for foliated manifolds.