arXiv research

Homogeneous three-spheres have only homogenous foliations.

Classifies polar foliations on symmetric spaces.

Lie foliations with symmetric leaves are smoothly conjugate to homogeneous ones.

The paper studies foliations on homogeneous spaces and identifies specific foliations.

Classifies foliations on specific symmetric spaces.

Non-unimodular foliations have specific geometric properties.

We complete the classification, initiated by the second named author, of homogeneous singular Riemannian foliations of spheres that are lifts of foliations produced from Clifford systems.

Study of flat Legendre transforms on complex projective plane.

We prove that, up to isometric congruence, there are exactly 2n+1 homogeneous polar foliations of the complex hyperbolic space. We also give an explicit description of each of these foliations.

A foliation on a Riemannian manifold is hyperpolar if it admits a flat section, that is, a connected closed flat submanifold that intersects each leaf of the foliation orthogonally. In this article we classify the hyperpolar homogeneous foliations on every Riemannian symmetric space of noncompact type.

Enhances Molino's description for foliated spaces, simplifying their study.

14 homogeneous convex foliations of degree 5 found on complex projective plane.

Introduces homogeneous G-structures to unify various geometric and foliation types.

Classification extended for reducible symmetric spaces.

New examples of isoparametric foliations on $S^n imes S^n$ are provided.

Homogeneous Einstein metrics on Euclidean spaces are shown to be Einstein solvmanifolds.

5 homogeneous convex foliations of degree 4 found on complex projective plane.

Study constant mean curvature tubes in homogeneous spaces.

Survey on CMC surfaces in homogeneous 3-manifolds.

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

The paper studies metrics with constant scalar curvature on foliated manifolds.

The study finds that certain curved manifolds can be mapped to symmetric spaces.

A foliation F on a Riemannian manifold M is homogeneous if its leaves coincide with the orbits of an isometric action on M. A foliation F is polar if it admits a section, that is, a connected closed totally geodesic submanifold of M which intersects each leaf of F, and intersects orthogonally at each point of intersect…

In this paper, we study stability for harmonic foliations on locally conformal Kähler manifolds with complex leaves. We also discuss instability for harmonic foliations on compact submanifolds immersed in Euclidean spaces and compact homogeneous spaces.

Let G be a simple Lie group of real rank one, and S the ideal boundary of the corresponding symmetric space of noncompact type (H^n_R, H^n_C, H^n_H or H^2_O). We show the finiteness of the possible values of the secondary characteristic classes of transversely homogeneous foliations on a fixed manifold whose transverse…

Classifies homogeneous hypersurfaces in specific 4D geometries.

For transversely homogeneous foliations on compact manifolds whose global holonomy group has connected closure, it is shown that either all holonomy covers of the leaves have polynomial growth with degree bounded by a common constant, or all holonomy covers of the leaves have exponential growth. This is an extension of…

Classifies Lie algebras actions on 3D spaces, focusing on solvable groups.

The paper proves a theorem about completely solvable Lie foliations on compact manifolds.

Study invariant Poisson structures on homogeneous manifolds, algebraically and geometrically.

We study holomorphic foliations with an affine homogeneous transverse structure. We give a friendly characterization of the case of transversely affine foliations in terms of matrix valued pairs of differential forms. This leads naturally to the study of the case of foliations with singularities. A first extension theo…

Classifies actions on complex space forms with Lagrangian orbits.

Study shows algebraic nature of manifold submetries on compact spaces.

Classifies polar actions on 3D homogeneous spaces.

Poisson homogeneous spaces for Poisson groupoids are classfied in terms of Dirac structures for the corresponding Lie bialgebroids. Applications include Drinfel'd's classification in the case of Poisson groups and a description of leaf spaces of foliations as homogeneous spaces of pair groupoids.

The paper shows measures equidistribute on affine submanifolds with a rate.

The purpose of this paper is to establish a completely new partial regularity theory on certain homogeneous complex Monge-Ampere equations. Our partial regularity theory will be obtained by studying foliations by holomorphic curves and and their relations to homogeneous complex Monge-Ampere equations. As applications, …

We study the problem of the existence and the holomorphicity of the Monge-Ampère foliation associated to a plurisubharmonic solutions of the complex homogeneous Monge-Ampère equation even at points of arbitrary degeneracy. We obtain good results for real analytic unbounded solutions. As a consequence we also provide a …

The study establishes a criterion for the holomorphy of curvature in smooth webs and applies it to dual webs of homogeneous foliations.

The Gromoll-Meyer sphere is constructed and studied in a homogeneous space.

We consider strict and complete nearly Kaehler manifolds with the canonical Hermitian connection. The holonomy representation of the canonical Hermitian connection is studied. We show that a strict and complete nearly Kaehler is locally a Riemannian product of homogenous nearly Kaehler spaces, twistor spaces over quate…

In this paper we study surfaces foliated by a uniparametric family of circles in the homogeneous space Sol$_3$. We prove that there do not exist such surfaces with zero mean curvature or with zero Gaussian curvature. We extend this study considering surfaces foliated by geodesics, equidistant lines or horocycles in tot…

Irreducible isoparametric foliations of arbitrary codimension q on complex projective spaces CP^n are classified, except if n=15 and q=1. Remarkably, there are noncongruent examples that pull back under the Hopf map to congruent foliations on the sphere. Moreover, there exist many inhomogeneous isoparametric foliations…

We classify irreducible polar foliations of codimension $q$ on quaternionic projective spaces $\mathbb H P^n$, for all $(n,q)\neq(7,1)$. We prove that all irreducible polar foliations of any codimension (resp. of codimension one) on $\mathbb H P^n$ are homogeneous if and only if $n+1$ is a prime number (resp. $n$ is ev…

Study of Lorentzian manifolds with specific transformations.

Geometric conditions are given so that the leafwise reduced cohomology is of infinite dimension, specially for foliations with dense leaves on closed manifolds. The main new definition involved is the intersection number of subfoliations with "appropriate coefficients". The leafwise reduced cohomology is also described…

We present a new method for manufacturing complex-valued harmonic morphisms from a wide class of Riemannian Lie groups. This yields new solutions from an important family of homogeneous Hadamard manifolds. We also give a new method for constructing left-invariant foliations on a large class of Lie groups producing harm…

Proves a principle for one-phase Bernoulli problem minimizers.