arXiv research

The study limits the number of specific foliations with bounded geometry.

Study on complex tori foliations and flat geometries.

Survey on holomorphic structures on complex manifolds.

Global existence and geometry of constant mass aspect function foliation in perturbed Schwarzschild spacetime studied.

We review basic notions and methods of noncommutative geometry and their applications to analysis and geometry on foliated manifolds.

A leafwise Hodge decomposition was proved by Sanguiao for Riemannian foliations of bounded geometry. Its proof is explained again in terms of our study of bounded geometry for Riemannian foliations. It is used to associate smoothing operators to foliated flows, and describe their Schwartz kernels. All of this is extend…

We review some applications of noncommutative geometry to the study of transverse geometry of Riemannian foliations and discuss open problems.

The study proves a transverse diameter theorem for Lorentzian foliations.

Study natural foliations in cotangent bundles of Cartan spaces.

We construct spectral triples in a sense of noncommutative differential geometry, associated with a Riemannian foliation on a compact manifold, and describe its dimension spectrum.

We use the CR geometry of the standard hyperquadric in complex projective three-space to give a detailed twistor description of conformal foliations in Euclidean three-space.

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

We study spectral asymptotics for the Laplace operator on differential forms on a Riemannian foliated manifold equipped with a bundle-like metric in the case when the metric is blown up in directions normal to the leaves of the foliation. The asymptotical formula for the eigenvalue distribution function is obtained. Th…

New foliations constructed from contact pairs, revealing flexible taut foliations.

A rigid submanifold result in contact geometry.

Geometric framework for inverse problems using foliations and dual connections.

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 chang…

We prove a theorem that gives a sufficient condition for the full basic automorphism group of a complete Cartan foliation to admit a unique (finite-dimensional) Lie group structure in the category of Cartan foliations. Emphasize that the transverse Cartan geometry may not be effective. Some estimates of the dimension o…

In this paper, we use the methods of subriemannian geometry to study the dual foliation of the singular Riemannian foliation induced by isometric Lie group actions on a complete Riemannian manifold M. We show that under some conditions, the dual foliation has only one leaf.

We discuss the geometry of warped foliations. After examining the Levi-Civita connection, we describe the formulae for sectional, Ricci and scalar curvatures. In the final part of this note, we present some examples.

This paper gives a survey of the index theory of tangentially elliptic and transversally elliptic operators on foliated manifolds as well as of related notions and results in non-commutative geometry.

Introduces a new equivalence for singular foliations and their groupoids.

First, we survey some results on classical and quantum dynamical systems associated with transverse Dirac operators on Riemannian foliations. Then we illustrate these results by two examples of Riemannian foliations: a foliation given by the fibers of a fibration and a linear foliation on the two-dimensional torus.

We study the geometry of the leaf closure space of regular and singular Riemannian foliations. We give conditions which assure that this leaf space is a singular symplectic or Kähler space.

Characterizes flag geometries for Hitchin representations in SL3(R).

Study local invariants and geometry of sub-Laplacian on H-type foliations.

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

New theorem extends Hadamard's to transversely affine geometry.

Classifies holomorphic parabolic geometries on complex manifolds.

Symmetric Poisson structures linked to geodesic foliations and Jordan algebras.

Lecture notes on singular foliations, smooth and holomorphic.

The paper develops surgery theories for foliations and solves a problem posed by Weinberger.

The paper is focused on the existence problem of attractors for foliations. Since the existence of an attractor is a transversal property of the foliation, it is natural to consider foliations admitting transversal geometric structures. As transversal structures are chosen Cartan geometries due to their universality. T…

A family of probability distributions parametrized by an open domain $Λ$ in $R^n$ defines the Fisher information matrix on this domain which is positive semi-definite. In information geometry the standard assumption has been that the Fisher information matrix tensor is positive definite defining in this way a Riemannia…

We obtain geometric characterizations of isospectral minimal Riemannian Legendre foliations on compact Sasakian manifolds of constant $φ$-sectional curvature.

We show that a classical result of Gromov in symplectic geometry extends to the context of symplectic foliations, which we regard as a $h$-principle for (regular) Poisson geometry. Namely, we formulate a sufficient cohomological criterion for a regular bivector to be homotopic to a regular Poisson structure, in the spi…

Research shows conditional existence of foliations by CMC and Willmore type half-spheres near a boundary point.

The paper studies the geometry and topology of a specific foliation on a complex surface.

Study numerically flat foliations on Kähler manifolds, proving new splitting theorems.

The main result of the paper is Egorov's theorem for transversally elliptic operators on compact foliated manifolds. This theorem is applied to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.

We describe notions of tautness that arise in the study of $C^0$ foliations, $C^{1,0}$ or smoother foliations, and in geometry. We give examples to show that these notions are different, and discuss how these differences impact some classical foliation results. We construct examples of smoothly taut $C^{\infty,0}$ foli…

Constructs a Lie groupoid integrating singular foliations.

Any leafwise connection on a fibre bundle over a foliated manifold is proved to come from a connection on this fibre bundle.

Study cohomology of quaternionic foliations and orbifolds.

3-quasi-Sasakian manifolds were studied systematically by the authors in a recent paper as a suitable setting unifying 3-Sasakian and 3-cosymplectic geometries. This paper throws new light on their geometric structure which reveals to be generally richer compared to the 3-Sasakian subclass. In fact, it turns out that t…

Extrinsic Geometric Flow (EGF) for a codimension-one foliation has been recently introduced by authors as deformations of Riemannian metrics subject to quantities expressed in terms of its second fundamental form. In the paper we introduce soliton solutions to EGF and study their geometry for totally umbilical foliatio…

Classifies homogeneous hypersurfaces in specific 4D geometries.

The study examines conditions that prevent null geodesic lines in spacetimes, impacting cosmological geometry.