arXiv research

Paper corrects and expands stochastic Lie systems theory.

Foliate systems are those which preserve some (possibly singular) foliation of phase space, such as systems with integrals, systems with continuous symmetries, and skew product systems. We study numerical integrators which also preserve the foliation. The case in which the foliation is given by the orbits of an action …

Investigates integrable systems with linear periodic integral for e(3) Lie algebra.

Study derived Lie ∞-groupoids and algebroids in higher differential geometry.

The study analyzes stochastic Lie systems and their applications in various models.

New examples of rigid Lie foliations with dense leaves found.

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

Study rigid Lie affine foliations on compact manifolds.

This paper proves a version for stochastic differential equations of the Lie-Scheffers Theorem. This result characterizes the existence of nonlinear superposition rules for the general solution of those equations in terms of the involution properties of the distribution generated by the vector fields that define it. Wh…

In this paper we try to generalize the Haefliger theorem on completly solvable Lie foliations. We prove that: every completely solvable Lie foliation on a compact manifold is the inverse image of a homogenus foliation. Every manifold in this paper is compact and our Lie group G is connexe and simply connexe.

In this work, we study Lie groupoids equipped with multiplicative foliations and the corresponding infinitesimal data. We determine the infinitesimal counterpart of a multiplicative foliation in terms of its core and sides together with a partial connection satisfying special properties, giving rise to the concept of I…

Establishes necessary and sufficient conditions for smooth triviality of Lie subalgebras and Lie ideals, and proves Moser's trick for foliations.

Study conformal foliations on Lie groups, finding new families and harmonic morphisms.

Study of centralizer elements preserving geodesic flow foliations on covers.

The study classifies nilpotent Lie foliations with cohomological obstructions.

New families of Lie groups with special foliations discovered.

The paper extends Riemann-Hilbert correspondence to foliations.

This paper extends foliation concepts to singular foliations using Lie $\infty$-algebroids.

Study on Lie groups' conformal foliations and harmonic morphisms.

New Lie groupoid and algebroid constructed for octonionic Hopf foliation.

Study the pullbacks and blowups of Lie algebroids and related structures.

Proves a Thom isomorphism for foliated differential forms.

If $A$ is a Lie algebroid over a foliated manifold $(M,\mathcal{F})$, a foliation of $A$ is a Lie subalgebroid $B$ with anchor image $T\mathcal{F}$ and such that $A/B$ is locally equivalent with Lie algebroids over the slice manifolds of $\mathcal{F}$. We give several examples and, for foliated Lie algebroids, we discu…

Paper studies symmetries in singular foliations using Lie $\infty$-morphisms.

The paper classifies foliations formed by generic coadjoint orbits of specific Lie groups.

The purpose of this paper is to show that any extension of a minimal Lie foliation on a compact manifold is a transversaly Riemannian g\h- foliation with trivial normal bundle. This result permits to classify the extensions of a minimal Lie foliation on a compact manifold from the Lie subgroups of its Lie group.

In this paper we study the Lie groupoids which appear in foliation theory. A foliation groupoid is a Lie groupoid which integrates a foliation, or, equivalently, whose anchor map is injective. The first theorem shows that, for a Lie groupoid G, the following are equivalent: - G is a foliation groupoid, - G has discrete…

The abstract describes a foliation of orbits for a specific class of Lie groups.

We associate a Lie $\infty$-algebroid to every resolution of a singular foliation, where we consider a singular foliation as a locally generated $\mathscr{O}$-submodule of vector fields on the underlying manifold closed under Lie bracket. Here $\mathscr{O}$ can be the ring of smooth, holomorphic, or real analytic funct…

A singular (or Hermann) foliation on a smooth manifold $M$ can be seen as a subsheaf of the sheaf $\mathfrak{X}$ of vector fields on $M$. We show that if this singular foliation admits a resolution (in the sense of sheaves) consisting of sections of a graded vector bundle of finite type, then one can lift the Lie brack…

The paper studies Morse theory for Lie algebra actions on Riemannian foliations.

Decomposes flows with jumps into simpler components.

We combine classic stability results for foliations with recent results on deformations of Lie groupoids and Lie algebroids to provide a cohomological characterization for rigidity of compact foliations on compact manifolds.

We describe a local model for any Singular Riemannian Foliation in a neighbourhood of a closed saturated submanifold of a regular stratum. Moreover we construct a Lie groupoid which controls the transverse geometry of the linear approximation of the Singular Riemannian Foliation around these submanifolds. We also discu…

This research extends Lie algebra actions to singular foliations.

Constructs a Lie groupoid integrating singular foliations.

Minimal conformal foliations on Lie groups are shown to be fibres of harmonic morphisms.

The study classifies natural almost Hermitian structures on specific Lie groups.

Generalizes energy-momentum method for non-autonomous Hamiltonian systems.

Higher homotopy generalizations of Lie-Rinehart algebras, Gerstenhaber-, and Batalin-Vilkovisky algebras are explored. These are defined in terms of various antisymmetric bilinear operations satisfying weakened versions of the Jacobi identity, as well as in terms of operations involving more than two variables of the L…

In this paper we study (smooth and holomorphic) foliations which are invariant under transverse actions of Lie groups.

Computes $L_\infty$-algebroid for linear foliations on vector spaces.

Motivated by questions of deformations/moduli in foliation theory, we investigate the structure of some groups of diffeomorphisms preserving a foliation. We give an example of a $C^\infty$ foliation whose diffeomorphism group is not a Lie group in any reasonable sense. On the positive side, we prove that the automorphi…

We study Lie foliations on compact manifolds, in case the Lie group is compact. Our main results improve Tischler classical result on the existence of fibration and, as an application, we study the case the manifold has an amenable fundamental group.

A pre-Lie algebroid is an anchored bundle provided with an almost Lie bracket such that the anchor is compatible with the Lie bracket of vector fields. We firstly show how most geometrical structures intensively studied in the framework of Lie algebroid can easily be extended in the pre-Lie algebroid context. The princ…

Study topological properties of integrable case on Lie algebra so(4).

This thesis treats two main topics: calibrated symplectic foliations, and local Lie groupoids. Calibrated symplectic foliations are one possible generalization of taut foliations of 3-manifolds to higher dimensions. Their study has been popular in recent years, and we collect several interesting results. We then show h…

It is well-known that a Lie algebroid A is equivalently described by a degree 1 Q-manifold M. We study distributions on M, giving a characterization in terms of A. We show that involutive Q-invariant distributions on M correspond bijectively to IM-foliations on A (the infinitesimal version of Mackenzie's ideal systems)…