arXiv research

The horizontal Laplacian of a Riemannian submersion with totally geodesic fibers and an integrable horizontal distribution.

We show a higher order integrability theorem for distributions generated by a family of vector fields under a horizontal regularity assumption on their coefficients. We use as chart a class of almost exponential maps which we discuss in details

Involutivity is a well known necessary condition for integrability of smooth tangent distributions. We show that this condition is still necessary for integrability with Sobolev surfaces. We specialize our study to the left invariant horizontal distribution of the first Heisenberg group $\H^1$. Here we answer a questio…

The paper derives inequalities for Riemannian maps and submersions involving quaternionic space forms.

This paper presents an investigation of the relation between some positivity of the curvature and the finiteness of fundamental groups in semi-Riemannian geometry. We consider semi-Riemannian submersions $π: (E, g) \rightarrow (B, -g_{B})$ under the condition with $(B, g_{B})$ Riemannian, the fiber closed Riemannian, …

We prove several versions of Driver's integration by parts formula for the horizontal Wiener measure on a totally geodesic Riemannian foliation and prove that the horizontal Wiener measure has a quasi-invariance property with respect to flows generated by suitable tangent processes.

Given a $C^1$ planes distribution $P_T$ on all ${\mathbb R}^m$ we consider {\em horizontal $α$-harmonic maps}, $α\ge 1/2$, with respect to such a distribution. These are maps $u\in H^α({\mathbb R}^k,{\mathbb R}^m)$ satisfying $P_T\nabla u=\nabla u$ and $P_T(u)(-Δ)^αu=0$ in ${\mathcal D}'({\mathbb R}^k).$ If the distrib…

This work is a short, self-contained introduction to subriemannian geometry with special emphasis on Chow's Theorem. As an application, a regularity result for the Poincaré Lemma is presented. At the beginning, the definitions of a subriemannian geometry, horizontal vector fields and horizontal curves are given. Then t…

Defines conformal submersion with horizontal distribution and provides necessary conditions for its existence.

Study on curvatures of surfaces in specific Lie groups.

We investigate certain natural connections between subriemannian geometry and hyperbolic dynamical systems. In particular, we study dynamically defined horizontal distributions which split into two integrable ones and ask: how is the energy of a subriemannian geodesic shared between its projections onto the integrable …

Study horizontal discs in fat distributions, proving their existence.

Marden and Strebel established the Heights Theorem for integrable holomorphic quadratic differentials on parabolic Riemann surfaces. We extends the validity of the Heights Theorem to all surfaces whose fundamental group is of the first kind. In fact, we establish a more general result: the {\it horizontal} map which as…

The purpose of this paper is to study anti-invariant Riemannian submersions from Kenmotsu manifolds onto Riemannian manifolds. Several fundamental results in this respect are proved. The integrability of the distributions and the geometry of foliations are investigated. We proved that there do not exist (anti-invariant…

Using the notion of Levi form of a smooth distribution, we discuss the local and the global problem of existence of one horizontal section of a smooth vector bundle endowed with a horizontal distribution. The analysis will lead to the formulation of a "one-leaf" analogue of the classical Frobenius integrability theorem…

In this paper, we study Riemannian, anti-invariant Riemannian and Lagrangian submersions. We prove that the horizontal distribution of a Lagrangian submersion from a Kaehlerian manifold is integrable. We also give some applications of this result. Moreover, we investigate the effect of the submersion to the geometry of…

We study 4-dimensional Riemannian manifolds equipped with a minimal and conformal foliation $\mathcal F$ of codimension 2. We prove that the two adapted almost Hermitian structures $J_1$ and $J_2$ are both cosymplectic if and only if $\mathcal F$ is Riemannian and its horizontal distribution $\mathcal H$ is integrable.

We show that, for an affine submersion $π: \mathbf{M}\longrightarrow \mathbf{B}$ with horizontal distribution, $\mathbf{B}$ is a statistical manifold with the metric and connection induced from the statistical manifold $\mathbf{M}$. The concept of conformal submersion with horizontal distribution is introduced, which i…

In this paper, we will first derive a DDVV-type optimal inequality for real skew-symmetric matrices, then we apply it to establish a Simons-type integral inequality for Riemannian submersions with totally geodesic fibres and Yang-Mills horizontal distributions. In this way, we show phenomenons of duality between Subman…

In the present paper we consider generic Sub-Riemannian structures on the co-rank 1 non-holonomic vector distributions and introduce the associated canonical volume and ''horizontal'' area forms. As in the classical case, the Sub-Riemannian minimal surfaces can be defined as the critical points of the '`horizontal'' ar…

The article proves the existence of horizontal immersions into fat distributions and contact structures.

New model for STSs with restricted horizontal gluings, focusing on maximal horizontal cylinders.

Whereas subriemannian geometry usually deals with smooth horizontal distributions, partially hyperbolic dynamical systems provide many examples of subriemannian geometries defined by non-smooth (namely, Hölder continuous) distributions. These distributions are of great significance for the behavior of the parent dynami…

Study of CR-submanifolds in various Lorentzian manifolds.

We study maximal horizontal subgroups of Carnot groups of Heisenberg type. We classify those of dimension half of that of the canonical distribution ("lagrangians") and illustrate some notable ones of small dimension. An infinitesimal classification of the arbitrary maximal horizontal submanifolds follows as a conseque…

The purpose of the present work is to study the complete and horizontal lifts of the metallic structure on tangent bundles with respect to almost product structure. We also establish fundamental formulae related to integrability and horizontal lifts of metallic structures on tangent bundles. Moreover, the study reveale…

Horizontal points of smooth submanifolds in stratified groups play the role of singular points with respect to the Carnot-Carathe'odory distance. When we consider hypersurfaces, they coincide with the well known characteristic points. In two step groups, we obtain pointwise estimates for the Riemannian surface measure …

The study characterizes infinite Riemann surfaces and their foliations using quadratic differentials.

In this paper, we consider a smooth connected finite-dimensional manifold $M$, an affine connection $\nabla$ with holonomy group $H^{\nabla}$ and $Δ$ a smooth completely non integrable distribution. We define the $Δ$-horizontal holonomy group $H^{\;\nabla}_Δ$ as the subgroup of $H^{\nabla}$ obtained by $\nabla$-paralle…

We consider coefficient bodies $\mathcal M_n$ for univalent functions. Based on the Löwner-Kufarev parametric representation we get a partially integrable Hamiltonian system in which the first integrals are Kirillov's operators for a representation of the Virasoro algebra. Then $\mathcal M_n$ are defined as sub-Riemann…

We introduce the notions of h-conformal slant submersions and almost h-conformal slant submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, slant submersions, h-slant submersions, almost h-slant submersion a…

Examples of area-minimizing graphs with low regularity in a specific group.

Study of contact-complex Riemannian submersions in various manifold classes.

We prove that the horizontal and vertical distributions of the tangent bundle with the Sasaki metric are isocline, the distributions given by the kernels of the horizontal and vertical lifts of the contact form $ω$ from the Heisenberg manifold $(H_3,g)$ to $(TH_3,g^S)$ are not totally geodesic, and the distributions $F…

The paper solves the Integration Problem for principal connections.

Complete integrability proved for SR geodesic flow on S^7.

By studying the group of rigid motions, $PSH(1)$, in the 3D-Heisenberg group $H_1$, we define the density and the measure for the sets of horizontal lines. We show that the volume of a convex domain $D\subset H_1$ is equal to the integral of length of chord over all horizontal lines intersecting $D$. As the classical r…

We study the uniqueness of generalized $p$-minimal surfaces in the Heisenberg group. The generalized $p$-area of a graph defined by $u$ reads $\int |\nabla u+\vec{F}| + Hu$. If $u$ and $v$ are two minimizers for the generalized $p$-area satisfying the same Dirichlet boundary condition, then we can only get $N_{\vec{F}}…

Paper extends Noether's Theorem to nonholonomic systems, proving conserved momentum.

We consider examples of the $\mathbb H$-type groups with the natural horizontal distribution generated by the commutation relations of the group. In the contrast with the previous studies we furnish the horizontal distribution with the Lorentzian metric, which is nondegenerate metric of index 1 instead of a positive de…

We prove a family of new Weitzenböck formulas on a Riemannian foliation with totally geodesic leaves. These Weitzenböck formulas are naturally parametrized by the canonical variation of the metric. As a consequence, under natural geometric conditions, the horizontal Laplacian satisfies a generalized curvature dimension…

Consider $L$ a regular Lagrangian, $S$ the canonical semispray, and $h$ the horizontal projector of the canonical nonlinear connection. We prove that if the Lagrangian is constant along the integral curves of the Euler-Lagrange equations then it is constant along the horizontal curves of the canonical nonlinear connect…

We study pseudoholomorphic curves in the nearly Kalher $\mathbf{CP}^3$. It is shown that a class of curves called null-torsion are in one to one correspondence with the integrals of a holomorphic contact system on the usual Kahler $\mathbb{CP}^3$ studied by Bryant. Browing Bryant's result we get plenty of such curves. …

It is known that there are surface bundles of arbitrarily high genus which have genus two Heegaard splittings. The simplest examples are Seifert fibered spaces with the sphere as a base space, three exceptional fibers and which allow horizontal surfaces. We characterize the monodromy maps of all surface bundles with ge…

Analytic curves have infinite codimension of singular germs.

Constructs covariant derivatives for Ehresmann connections.

The excessively increased volume of data in modern data management systems demands an improved system performance, frequently provided by data distribution, system scalability and performance optimization techniques. Optimized horizontal data partitioning has a significant influence of distributed data management syste…

In this paper, we define almost paracontact and normal almost paracontact Finsler structures on a vector bundle and find some conditions for integrability of these structures. We define paracontact metric, para- Sasakian and K-paracontact Finsler structures and study some properties of these structures. For a K-paracon…