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Valera Berestovskii

Publications and source records attributed to Valera Berestovskii.

11 recordsLinked to original sources
arXiv2023

Sub-Lorentzian geodesics on ${\rm GL}^{+}(2,\mathbb{C})$ with the generating space of Hermitian matrices in the Lie algebra $\mathfrak{gl}^{+}(2,\mathbb{C})$

The Lie subgroup ${\rm GL}^{+}(2,\mathbb{C})$ of all matrices in the Lie group ${\rm GL}(2,\mathbb{C})$ with positive real determinant is equipped with a left-invariant sub-Lorentzian (anti)metric defined by the natural structure of the 4-dimensional Minkowski space-time on the subspace of Hermitian matrices in its Lie algebra. In base of the corresponding time-anti-optimal control problem, formulated in the paper, and Pontryagin minimum principle for it, using geodesics and shortest arcs of the corresponding left-invariant sub-Riemannian metric on the Lie subgroup ${\rm SL}(2,\mathbb{C})$, the authors found sub-Lorentzian nonspacelike geodesics and longest arcs.

math.DG↗
arXiv2021

Abnormal extremals of left-invariant sub-Finsler quasimetrics on four-dimensional Lie groups with three-dimensional generating distributions

We find three-dimensional subspaces of four-dimensional connected Lie algebras, generating these algebras, and abnormal extremals on connected Lie groups with these Lie algebras and with left-invariant sub-Finsler quasimetrics defined by seminorms on such subspaces. In terms of the structure constants of Lie algebras and dual seminorms, we establish a criterion for the strong abnormality of these extremals.

math.DG↗
arXiv2020

Abnormal extremals of left-invariant sub-Finsler quasimetrics on four-dimensional Lie groups

Abnormal extremals on four-dimensional connected Lie groups with left-invariant sub-Finsler quasimetric, defined by a seminorm on a two-dimensional subspace of the Lie algebra generating the algebra, are found. In terms of structure constant of Lie algebra and supporting Minkowski function for the unit ball of seminorm on two-dimensional subspace of Lie algebra, defining a quasimetric, we establish a criterion for strict abnormality of these extremals.

math.DG↗
arXiv2020

Extremals of a left-invariant sub-Finsler metric on the Engel group

The authors found extremals of arbitrary left-invariant sub-Finsler metric on the Engel group defined by a distribution of rank two. They use for this the Pontryagin Maximum Principle for the corresponding time-optimal problem in coordinates of the first kind. The obtained results are applied to the case of left-invariant sub-Riemannian metric with the same distribution.

math.DG↗
arXiv2019

Pontryagin maximum principle, (co)adjoint representation, and normal geodesics of left-invariant (sub-)Finsler metrics on Lie groups

On the ground of origins of the theory of Lie groups and Lie algebras, their (co)adjoint representations, and the Pontryagin maximum principle for the time-optimal problem are given an independent foundation for methods of geodesic vector field to search for normal geodesics of left-invariant (sub-)Finsler metrics on Lie groups and to look for the corresponding locally optimal controls in (sub-)Riemannian case, as well as some their applications.

math.DG↗
arXiv2006

Generalized Universal Covers of Uniform Spaces

We develop a generalized covering space theory for a class of uniform spaces called coverable spaces. Coverable spaces include all geodesic metric spaces, connected and locally pathwise connected compact topological spaces, in particular Peano continua, as well as more pathological spaces like the topologist's sine curve. Each coverable space has a generalized universal cover with universal and lifting properties. Associated with this generalized universal cover is a functorial uniform space invariant called the deck group, which is related to the classical fundamental group by a natural homomorphism. We obtain some specific results for one-dimensional spaces. Keywords: universal cover, uniform space, geodesic space, fundamental group

math.AT↗