Searcharxiv⌕ Search

arXiv subjects

S. N. Stelmastchuk

Publications and source records attributed to S. N. Stelmastchuk.

12 recordsLinked to original sources
arXiv2026

Algebraic Conjugacy of Linear Flows on Connected Lie Groups

We study algebraic conjugacy of linear flows on connected Lie groups. To each linear flow we associate a derivation of the Lie algebra, and we analyze how the conjugacy problem can be transferred to the infinitesimal level. In the connected and simply connected case, algebraic conjugacy is characterized by the conjugacy of the associated derivations. For non-simply connected Lie groups, we show that this condition must be supplemented by global restrictions arising from the universal covering group. More precisely, only projectable derivations define linear flows on the quotient, and only admissible Lie algebra isomorphisms descend to conjugacies between the quotient groups. This provides a framework for the algebraic classification of linear flows by computing projectable derivations and admissible automorphisms. We also describe fixed point subgroups and kernels of derivations as obstructions to algebraic conjugacy. Finally, we apply the general results to the three-dimensional Heisenberg group and to its central quotients, showing explicitly how the quotient topology restricts both the admissible derivations and the conjugating automorphisms.

math.DS↗
arXiv2026

Projective Controllability of Complete Lifted Control Systems

We introduce the complete lifted control system associated with a control system on a smooth manifold by replacing each vector field with its complete lift to the tangent bundle. We prove that complete lifted control systems are never controllable on the whole tangent bundle, due to the invariance of the zero section. Motivated by this obstruction and by the invariance of complete lifts under fiberwise dilations, we study the induced control system on the projectivized tangent bundle. We establish the relationship between the controllability properties of the lifted and projectivized systems, showing in particular that projective controllability implies controllability of the original system. Our main result provides a sufficient condition for controllability of the projectivized system in terms of a Lie rank condition modulo the Euler vector field.

math.OC↗
arXiv2026

Controllability of Linear Control Systems on Solvable Lie Groups with Hyperbolic Drift

We study the controllability of linear control systems on connected solvable Lie groups with hyperbolic drift. Under a suitable splitting assumption on the control directions, we show that the controllability problem can be reduced to the controllability of induced systems on the positive and negative hyperbolic components of the group. We then establish sufficient conditions ensuring controllability of these induced systems based on the ad-rank condition and suitable drift-reachability assumptions. As a consequence, we obtain a controllability criterion for the original system. The results provide a hyperbolic counterpart to existing controllability results for linear systems on solvable Lie groups and are illustrated by explicit examples.

math.OC↗
arXiv2026

Vertical Control Systems on Tangent Bundles and Fiberwise Controllability

We study control systems on the tangent bundle of a smooth manifold induced by vertical lifts of vector fields. The Vertical dynamics acts exclusively along the fibers, leaving the base point unchanged and reducing the system to a linear control problem on each tangent space, for which we obtain explicit solutions and characterize reachable sets, showing that fiberwise controllability is equivalent to a rank condition on the original vector fields. We then consider lifted systems combining complete drift and vertical controls, where the base trajectory is fixed by the drift and the control acts on tangent directions. For these systems, we derive explicit solutions and a complete characterization of reachable sets via a transport operator, yielding a necessary and sufficient condition for fiberwise controllability in terms of transported vector fields, together with a Lie-algebraic sufficient criterion.

math.OC↗
arXiv2021

Periodic orbits of Linear and invariant flows on connected Lie groups

Our main is to study periodic orbits of linear and invariant flows on a real, connected Lie group. Since each linear flow $φ_t$ has a derivation associated $\mathcal{D}$, we show that the existence of periodic orbits of $φ_t$ is based on the eigenvalues of the derivation $\mathcal{D}$. From this, we study periodic orbits of a linear flow on noncompact, semisimple Lie groups, and we work with periodic orbits of a linear flow on connected, simply connected, solvable Lie groups of dimension 2 or 3.

math.DS↗
arXiv2019

Linear flows on compact, semisimple Lie groups: stability, periodic orbits, and Poincaré-Bendixon's Theorem

Our first purpose is to study the stability of linear flows on real, connected, compact, semisimple Lie groups. After, we study and classify periodic orbits of linear and invariant flows. In particular, we obtain a version of Poincaré-Bendixon's Theorem. As an application, we present periodic orbits of linear or invariant flows on $SO(3)$ or $SU(2)$, and we classify periodic orbits of a linear or invariant system on $SO(4)$.

math.DS↗
arXiv2014

Martingales in Reductive Homogeneous spaces

The subject of this work is to study martingales in a reductive homogeneous space with respect to a symmetric connection. Our basic idea is to view homogenous spaces as principal fiber bundles and, thus, to study martingales on homogeneous space with aid of horizontal martingales on Lie group. Furthermore, using the stochastic logarithm we give a characterization of martingales on homogenous space. To end, we study the martingales in spheres $S^{n}$ and $SL(n,\mathbb{R})/SO(n,\mathbb{R})$, $n \geq 2$.

math.PR↗
arXiv2010

A characterization of Einstein manifolds

In this work we wish characterize the Einstein manifolds $(M,g)$, however without the necessity of hypothesis of compactness over $M$ and unitary volume of $g$, which are well known in many works. Our result says that if all eingenvalues $λ$ of $r_{g}$, with respect to $g$, satisfy $λ\geq \frac{1}{n}s_{g}$, then $(M,g)$ is an Einstein manifold, where $r_{g}$ and $s_{g}$ denote the Ricci and scalar curvatures, respectively.

math.DG↗
arXiv2010

Stochastic characterization of harmonic sections and a Liouville theorem

Let $P(M,G)$ be a principal fiber bundle and $E(M,N,G,P)$ be an associate fiber bundle. Our interested is to study harmonic sections of the projection $π_{E}$ of $E$ into $M$. Our first purpose is to give a stochastic characterization of harmonic section from $M$ into $E$ and a geometric characterization of harmonic sections with respect to its equivariant lift. The second purpose is to show a version of Liouville theorem for harmonic sections and to prove that section $M$ into $E$ is a harmonic section if and only if it is parallel.

math.DG↗
arXiv2009

A Pluzhnikov's Theorem, Brownian motions and Martingales in Lie Group with skew-symmetric connections

Let $G$ be a Lie Group with a left invariant connection such that its connection function is skew-symmetric. Our main goal is to show a version of Pluzhnikov's Theorem for this kind of connection. To this end, we use the stochastic logarithm. More exactly, the stochastic logarithm gives characterizations for Brownian motions and Martingales in $G$, and these characterzations are used to prove Pluzhnikov's Theorem.

math.DG↗
arXiv2009

An equivalence between harmonic sections and sections that are harmonic maps

Let $π:(E,\nabla^{E}) \to (M,g)$ be an affine submersion with horizontal distribution, where $\nabla^{E}$ is a symmetric connection and $M$ is a Riemannian manifold. Let $σ$ be a section of $π$, namely, $π\circ σ= Id_{M}$. It is possible to study the harmonic property of section $σ$ in two ways. First, we see $σ$ as a harmonic map. Second, we see $σ$ as harmonic section. In the Riemannian context, it means that $σ$ is a critical point of the vertical functional energy. Our main goal is to find conditions to the assertion: $σ$ is a harmonic map if and only if $σ$ is a harmonic section.

math.DG↗