Geometric resolution of singular Riemannian foliations

We prove that an isometric action of a Lie group on a Riemannian manifold admits a resolution preserving the transverse geometry if and only if the action is infinitesimally polar. We provide applications concerning topological simplicity of several classes of isometric actions, including polar and variationally complete ones. All results are proven in the more general case of singular Riemannian foliations.     

Parallel focal structure and singular Riemannian foliations

We give a necessary and sufficient condition for a submanifold with parallel focal structure to give rise to a global foliation of the ambient space by parallel and focal manifolds. We show that this is a singular Riemannian foliation with complete orthogonal transversals. For this object we construct an action on the transversals that generalizes the Weyl group action for polar actions.

     

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 Molino?s conjecture, are presented.     

Basic forms for transversely integrable singular Riemannian foliations

Basic forms for a transversely integrable singular Riemannian foliation with compact leaves are in one-to-one correspondence with ``Weyl"-invariant differential forms on a generalized section of the foliation.

     

Lie groupoids and semi-local models of singular Riemannian foliations

Annals of Global Analysis and Geometry - We describe a local model for any singular Riemannian foliation in a neighborhood of a closed saturated submanifold of a singular stratum. Moreover, we...     

Point leaf maximal singular Riemannian foliations in positive curvature

We generalize the notion of fixed point homogeneous isometric group actions to the context of singular Riemannian foliations. We find that in some cases, positively curved manifolds admitting these so-called point leaf maximal SRF's are diffeo/homeomorphic to compact rank one symmetric spaces. In all cases, manifolds admitting such foliations are cohomology CROSSes or finite quotients of them. Among non-simply connected manifolds, we find examples of such foliations which are non-homogeneous.     

Clifford algebras and new singular Riemannian foliations in spheres

Using representations of Clifford algebras we construct indecomposable singular Riemannian foliations on round spheres, most of which are non-homogeneous. This generalises the construction of non-homogeneous isoparametric hypersurfaces due to by Ferus, Karcher and Münzner.     

On mean curvature flow of singular Riemannian foliations: Noncompact cases

In this paper we investigate the mean curvature flow (MCF) of a regular leaf of a closed generalized isoparametric foliation as initial datum, generalizing previous results of Radeschi and the first author. We show that, under bounded curvature conditions, any finite time singularity is a singular leaf, and the singularity is of type I. The new techniques also allow us to discuss the existence of basins of attraction, how cylinder structures can affect convergence of basic MCF of immersed submanifolds and assure convergence of MCF of non-closed leaves of generalized isoparametric foliation on compact manifold.     

Totally geodesic foliations close to Riemannian foliations

The relation of the curvature and topology of totally geodesic foliations close to Riemannian ones is studied. The main result complements Ferus's famous theorem on totally geodesic foliations.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 114–118, 1992.     

Spectral geometry for Riemannian foliations

Let F be a Riemannian foliation on a Riemannian manifold (M, g), with bundle-like metric g. Aside from the Laplacian △_g associated to the metric g, there is another differential operator, the Jacobi operator J▽, which is a second order elliptic operator acting on sections of the normal bundle. Its spectrum is discrete as a consequence of the compactness of M. Hence one has two spectra, spec (M, g) = spectrum of △_g (acting on functions), and spec (F, J▽) = spectrum of J▽. We discuss the following problem: Which geometric properties of a Riemannian foliation F on a Riemannian manifold (M, g) are determined by the two types of spectral invariants?     

Minimizability of developable Riemannian foliations

Let (M,F){(M,\mathcal{F})} be a closed manifold with a Riemannian foliation. We show that the secondary characteristic classes of the Molino’s commuting sheaf of (M,F){(M,\mathcal{F})} vanish if (M,F){(M,\mathcal{F})} is developable and π ₁ M is of polynomial growth. By theorems of álvarez López in (álvarez López, Ann. Global Anal. Geom., 10:179–194, 1992) and (álvarez López, Ann. Pol. Math., 64:253–265, 1996), our result implies that (M,F){(M,\mathcal{F})} is minimizable under the same conditions. As a corollary, we show that (M,F){(M,\mathcal{F})} is minimizable if F{\mathcal{F}} is of codimension 2 and π ₁ M is of polynomial growth.     

Riemannian foliations of bounded geometry

Continuing the study of bounded geometry for Riemannian foliations, begun by Sanguiao, we introduce a chart‐free definition of this concept. Our main theorem states that it is equivalent to a condition involving certain normal foliation charts. For this type of charts, it is also shown that the derivatives of the changes of coordinates are uniformly bounded, and there are nice partitions of unity. Applications to a trace formula for foliated flows will be given in a forthcoming paper.     

Riemannian foliations admitting transversal conformal fields

Let be a closed, connected Riemannian manifold with a foliation of codimension q and a bundle-like metric g _M . We study the relationship between several infinitesimal automorphisms. Moreover under the some curvature condition, if M admits a transversal conformal field, then is transversally isometric to the action of a finite subgroup of O(q) acting on the q-sphere of constant curvature.      

Spectral sequences and taut Riemannian foliations

Using a relation between the terms of the spectral sequence of a Riemannian foliation and its adiabatic limit, we obtain Bochner type techniques for this special setting and, as a consequence, in the special case of a Riemannian flow we obtain vanishing conditions for the top dimensional group of the basic cohomology \(H_{b}^{q}(\mathcal{F})\)-which is related to the property of being geodesible. We also extend a Weitzenböck type formula for the leafwise Laplacian and, for the particular class of compact foliations, we obtain a generalization of a result due to Ph. Tondeur, M. Min-Oo, and E. Ruh concerning the vanishing of the basic cohomology under the assumption that certain curvature operators are positive definite. In the final part we present an example.