arXiv research

In this paper we prove the conjecture of Molino that for every singular Riemannian foliation $(M,\mathcal{F})$, the partition $\bar{\mathcal{F}}$ given by the closures of the leaves of $\mathcal{F}$ is again a singular Riemannian foliation.

We discuss the connection between the smooth and metric structure on quotient spaces, prove smoothness of isometries in special cases and discuss an application to a conjecture of Molino.

Investigates singular Finsler foliations on $(α,β)$-spaces and their relation to Riemannian foliations.

We prove here that given a proper isometric action $K\times M\to M$ on a complete Riemannian manifold $M$ then every continuous isometric flow on the orbit space $M/K$ is smooth, i.e., it is the projection of an $K$-equivariant smooth flow on the manifold $M$. As a direct corollary we infer the smoothness of isometric …

We complete the classification of compact connected contact toric manifolds initiated by Banyaga and Molino and by Galicki and Boyer. As an application we prove the conjectures of Toth and Zelditch on toric integrable systems on the n-torus and the 2-sphere.

Generalizes Molino's theory for Riemannian foliations.

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.

Inspired by the work of Molino, we show that the integrability obstruction for transitive Lie algebroids can be made to vanish by adding extra dimensions. In particular, we prove that the Weinstein groupoid of a non-integrable transitive and abelian Lie algebroid, is the quotient of a finite dimensional Lie groupoid. T…

Molino's description of Riemannian foliations on compact manifolds is generalized to the setting of compact equicontinuous foliated spaces, in the case where the leaves are dense. In particular, a structural local group is associated to such a foliated space. As an application, we obtain a partial generalization of res…

For a Riemannian foliation on a closed manifold, the first secondary invariant of Molino's central sheaf is an obstruction to tautness. Another obstruction is the class defined by the basic component of the mean curvature with respect to some metric. Both obstructions are proved to be the same up to a constant, and oth…

New method reconstructs hidden structures from noisy data.

Let $(M,ω)$ be a symplectic manifold endowed with a agrangian foliation ${\cal L}$, it has been shown by Weinstein [16] hat the symplectic structure of $M$ defines on each leaf of ${\cal L}$, connection which curvature and torsion forms vanish identically. uppose that $L_0$ is a compact leaf which Weinstein connection …

We prove that if the normal distribution of a singular riemannian foliation is integrable, then each leaf of this normal distribution can be extended to be a complete immersed totally geodesic submanifold (called section) which meets every leaf orthogonally. In addition the set of regular points is open and dense in ea…

In this paper, the Almeida-Molino obstruction to developability of transversely complete foliations is extended to Lie groupoids.

The topological Molino's description of equicontinuous foliated spaces, studied by the first author and Moreira Galicia, gives conditions to reduce their study to the particular case where the holonomy pseudogroup can be represented by a pseudogroup on some local group $G$ generated by some of its local left translatio…

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 the…

Study uses blow-up method to analyze foliations in Riemannian geometry.

From 1980s, it is an open problem of proposing cohomologic formula for the basic index of a transversally elliptic basic differential operator on a vector bundle over a foliated manifold. In 1990s, El Kacimi-Alaoui has proprosed to use the Molino theory for study this index. Molino has proved that to every transversall…

Survey on Killing foliations with technical advantages.

Let (M,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) vanish if (M,F) is developable and the fundamental group of M is of polynomial growth. By theorems of Álvarez López, our result implies that (M,F) is minimizable under t…

Examines respectful decompositions of Lie algebras.

Geometric framework for inverse problems using foliations and dual connections.

We show that any transversally complete Riemannian foliation F of dimension one on any possibly non-compact manifold M is tense; namely, (M,F) admits a Riemannian metric such that the mean curvature form of F is basic. This is a partial generalization of a result of Dominguez, which says that any Riemannian foliation o…

We show that the cone associated with a moment map for an action of a torus on a contact compact connected manifold is a convex polyhedral cone and that the moment map has connected fibers provided the dimension of the torus is bigger than 2 and that no orbit is tangent to the contact distribution. This may be consider…

Survey and extend work on singular foliations in diffeology.

Simplified proof of foliation closure theorem for linear foliations.

Consider a singular Riemannian foliation (s.r.f for short) on a compact manifold. By successive blow-ups along the strata, we construct a regular Riemannian foliation on another compact Riemannian manifold and a desingularization map that projects leaves of the regular Riemannian foliation into leaves of the s.r.f. Thi…

The abstract discusses transforming Vaisman metrics into Kähler-Einstein structures.

To a closed wide Lie subgroupoid $\mathbf{A}$ of a Lie groupoid $\mathbf{L}$, i.e. a Lie groupoid pair, we associate an Atiyah class which we interpret as the obstruction to the existence of $\mathbf{L}$-invariant fibrewise affine connections on the homogeneous space $\mathbf{L}/\mathbf{A}$. For Lie groupoid pairs with…

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$ or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the o…

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $SL(n+1, \mathbb{R})$ or $PGL(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the original ones. The …

We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a locally toric Lagrangian fibration. For a local torus action, we define two invariants…

The basic cohomology of a Riemannian foliation on a complete manifold with all leaves closed is the cohomology of the leaf space. In this paper we introduce various methods to compute the basic cohomology in the presence of both closed and non-closed leaves in the simply-connected case (or more generally for Killing fo…

There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three appr…

Polyhedra volume conjecture supports Stoker conjecture weakly.

Survey on two non-Kähler geometry conjectures.

Numerical study confirms Brennan's conjecture for a counterexample to Thurston's $K=2$ conjecture.

The non-vanishing conjecture implies the abundance conjecture in certain cases.

Proves Gromov's conjecture and answers Stoker's polyhedron conjecture.

This paper gives an algebraic conjecture which is shown to be equivalent to Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It generalizes the Stallings-Jaco theorem which established a similar result for the Poincare Conjecture. The paper also gives two other algebraic conjectures; one is equi…

Symmetry-breaking in three differential geometry conjectures.

We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particula…

Study proves Hecke lifting conjecture for torus knots and verifies it for any framed knots.

New proof shows most thin knots satisfy Cabling Conjecture.

The paper introduces SRFs and I-Poisson manifolds, linking foliations to Riemannian geometry.

We review the Burghelea conjecture, which constitutes a full computation of the periodic cyclic homology of complex group rings, and its relation to the algebraic Baum-Connes conjecture. The Burghelea conjecture implies the Bass conjecture. We state two conjectures about groups of finite asymptotic dimension, which tog…

Metric SYZ conjecture proved using non-archimedean geometry.

Paper confirms Chen's biharmonic conjecture for hypersurfaces in 5D.