arXiv subjects

Marek Grochowski

Publications and source records attributed to Marek Grochowski.

7 recordsLinked to original sources

arXiv2024

On the existence and prolongation of infinitesimal isometries on special sub-Riemannian manifolds

In the present paper we deal with (local) infinitesimal isometries of special sub-Riemannian manifolds (a contact and oriented sub-Riemannian manifold is called special if the Reeb vector field is an isometry). The objective of the paper is to find some conditions on such manifolds which allow one to construct, locally around a given point, infinitesimal isometries and then, possibly, to prolong them onto bigger domains. The mentioned conditions are related to the so-called $\is^*$-regular and $\is$-regular points, the notions introduced by Nomizu in [22] in the Riemannian setting and slightly modified by the author.

math.DG↗

arXiv2016

3-dimensional left-invariant sub-Lorentzian contact structures

We provide a classification of $ts$-invariant sub-Lorentzian structures on $3$ dimensional contact Lie groups. Our approach is based on invariants arising form the construction of a normal Cartan connection.

math.DG↗

arXiv2015

On contact sub-pseudo-Riemannian isometries

We study isometries in the contact sub-pseudo-Riemannian geometry. In particular we give an upper bound on the dimension of the isometry group of a general sub-pseudo-Riemannian manifold and prove that the maximal dimension is attained for the left invariant structures on the Heisenberg group.

math.DG↗

arXiv2015

Invariants and Infinitesimal Transformations for Contact Sub-Lorentzian Structures on 3-Dimensional Manifolds

In this article we develop some elementary aspects of a theory of symmetry in sub-Lorentzian geometry. First of all we construct invariants characterizing isometric classes of sub-Lorentzian contact 3 manifolds. Next we characterize vector fields which generate isometric and conformal symmetries in general sub-Lorentzian manifolds. We then focus attention back to the case where the underlying manifold is a contact 3 manifold and more specifically when the manifold is also a Lie group and the structure is left-invariant.

math.DG↗

arXiv2015

Invariants of contact sub-pseudo-Riemannian structures and Einstein-Weyl geometry

We consider local geometry of sub-pseudo-Riemannian structures on contact manifolds. We construct fundamental invariants of the structures and show that the structures give rise to Einstein-Weyl geometries in dimension 3, provided that certain additional conditions are satisfied.

math.DG↗

arXiv2013

A necessary and sufficient condition for local controllability around closed orbits

In this paper we give a necessary and sufficient condition for local controllability around closed orbits for general smooth control systems. We also prove that any such system on a compact manifold has a closed orbit.

math.OC↗

arXiv2013

Normal forms for sub-Lorentzian metrics supported on Engel type distributions

We construct normal forms for Lorentzian metrics on Engel distributions under the assumption that abnormal curves are timelike future directed Hamiltonian geodesics. Then we indicate some cases in which the abnormal timelike future directed curve initiating at the origin is geometrically optimal. We also give certain estimates for reachable sets from a point.

math.DG↗