On some classes of foliations

2017

In this work, we find an equation that relates the Ricci curvature of a riemannian manifold M and the second fundamental forms of two orthogonal foliations of complementary dimensions, F and F^, defined on M. Using this equation, we show a sufficient condition for the manifold M to be locally a riemannian product of the leaves of F and F^, if one of the foliations is totally umbilical. We also prove an integral formula for such foliations.

Mathematische Nachrichten, 2014

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 given in a forthcoming paper.

arXiv (Cornell University), 2021

An orbit-like foliation is a singular foliation on a complete Riemannian manifold M whose leaves are locally equidistant (i.e., a singular Riemannian foliation) and (transversely) infinitesimally homogenous. This class of singular foliation contains not only the classe of partion of the space into orbits of isometric actions, but also infinite many non homogenous examples and in particular the partition of M into orbits of a proper groupoid. In this paper we prove a version of Kondrakov Embedding Theorem and an analogous Principle of Symmetric Criticality of Palais for basic funcions of orbit-like foliations. As proof of concepts, we study not only the corresponding Yamabe problem in the setting, but also to the case of fiber bundles with homogeneous fibers, seeking for the existence of metrics with constant scalar curvature that respect the respective Riemannian Foliation decomposition. An application to the existence of positive constant scalar curvature on exotic spheres is presented. In an upcoming version we shall extend the results to the corresponding Kazdan-Warner problem.

We discuss main ingredients of (differential) tangential geometry of a given foliated manifold (M, F). The tangential geometry of (M, F) is essentially the (differential) geometry of the leafwise manifold M δ. A horizontal bundle of (M, F) is defined as a vector subbundle of the tangent bundle T M that is complementary to the vertical bundle of (M, F). Horizontal vector fields and horizontal sheets are naturally introduced. Fundamental vertical and/or tangential objects, like functions, vectors and forms are introduced and explored. Various characterizations of such objects are given. Finally, an important ingredient, namely the vertical frame bundle of the foliated manifold (M, F), is defined.

Publicacions Matemàtiques, 1994

Let .T be a singular Riemannian foliation on a compact connecte d Riemannian manifold M. We demonstrate that global foliate d vector fields generate a distribution tangent to the strata defined by the closures of leaves of .~ and which, in each stratum, is transverse to these closures of leaves. The aim of this short note is to prove M. Pierrot's theorem for singula r Riemanian foliations, cf. [5], namely. Theorem 1. Let F be an SRF on a compact manifold M. Then the vector space of global foliated vector fields is transitive to the closures of leaves in each closure stratum. 1. Preliminarie s First we recall some and prove other results about SRF-s (singular Riemannian foliations), cf. [3] and [4]. Assume that the manifold M is compact and connected (or the metri c is complete). Then the closure of any leaf is a submanifold. Let k be any number between o and n. Define E k ={xEM :xEdimL, = k}. The leaves of ,1 ' is Ek are of the same dimension, however they ca n have holonomy. P. Molino demonstrated that the sets E k or rather thei r connected components are submanifolds of M and C Ui<k Ei. Note that for some i the sets Ei can be empty. Moreover, let ko be the maximum dimension of leaves of F. Then the set E h is open and dense

2013

We show that a Riemannian foliation on a topological $n$-sphere has leaf dimension 1 or 3 unless n=15 and the Riemannian foliation is given by the fibers of a Riemannian submersion to an 8-dimensional sphere. This allows us to classify Riemannian foliations on round spheres up to metric congruence.

Differential Geometry and its Applications, 2009

Lagrangians related to submersions and foliations, which are analogous to Riemannian submersions and Riemannian foliations respectively are studied in the paper. One prove that a bundle-like Lagrangian, a transverse hyperregular Lagrangian, a hyperregular Lagrangian foliated cocycle or a geodesic orthogonal property are equivalent data for a foliation. A conjecture of E. Ghys is proved in a more general setting than that of Finslerian foliations: a foliation that has a transverse positively definite Lagrangian is a Riemannian foliation. One extend also a result of Miernowski and Mozgawa on Finslerian foliations, proving that the natural lift to the normal bundle of a Lagrangian foliation that has a transverse positively definite Lagrangian is a Riemannian foliation.

Russian Journal of Mathematical Physics, 2009

In this paper, we present new results on the tautness of Riemannian foliations in their historical context. The first part of the paper gives a short history of the problem. For a closed manifold, the tautness of a Riemannian foliation can be characterized cohomologically. We extend this cohomological characterization to a class of foliations which includes the foliated strata of any singular Riemannian foliation of a closed manifold.

Annales de l’institut Fourier, 1990

On riemannian foliations with minimal leaves Annales de l'institut Fourier, tome 40, n o 1 (1990), p. 163-176 <http://www.numdam.org/item?id=AIF_1990__40_1_163_0> © Annales de l'institut Fourier, 1990, tous droits réservés. L'accès aux archives de la revue « Annales de l'institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Manuscripta Mathematica, 2008

For a riemannian foliation \({\mathcal{F}}\) on a closed manifold M, it is known that \({\mathcal{F}}\) is taut (i.e. the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean curvature form \(\kappa_\mu\) (relatively to a suitable riemannian metric μ) is zero (cf. Álvarez in Ann Global Anal Geom 10:179–194, 1992). In the transversally orientable case, tautness is equivalent to the non-vanishing of the top basic cohomology group \(H^{n}\,(M\,/\,{\mathcal{F}})\) , where \(n = {\rm codim}\,{\mathcal{F}}\) (cf. Masa in Comment Math Helv 67:17–27, 1992). By the Poincaré Duality (cf. Kamber et and Tondeur in Astérisque 18:458–471, 1984) this last condition is equivalent to the non-vanishing of the basic twisted cohomology group \(H^{0}_{\kappa_\mu}(M\,/\,{\mathcal{F}})\) , when M is oriented. When M is not compact, the tautness class is not even defined in general. In this work, we recover the previous study and results for a particular case of riemannian foliations on non compact manifolds: the regular part of a singular riemannian foliation on a compact manifold (CERF).