arXiv subjects
Publications and source records attributed to Simão N. Stelmastchuk.
Let $P(M,G)$ be a principal fiber bundle, let $ω$ be a connection form on $P(M,G)$, and consider a projectable connection $\nabla^{P}$ on $P(M,G)$. The aim of this work is to determine the $\nabla^{P}$-martingales in $P(M,G)$. Our results allow establishing new characterizations of harmonic maps from Riemannian manifolds to principal fiber bundles.
The purpose of this paper is to describe explicitly the solution for linear control systems on Lie groups. In case of linear control systems with inner derivations, the solution is given basically by the product of the exponential of the associated invariant system and the exponential of the associated invariant drift field. We present the solutions in low dimensional cases and apply the results to obtain some controllability results.
The aim of this paper is to give a condition to topological conjugacy of invariant flows in an Lie group $G$ which its Lie algebra $\mathfrak{g}$ is associative algebra or semisimple. In fact, we show that if two dynamical system on $G$ are hyperbolic, then they are topological conjugate.
We factorize harmonic maps with values in a semisimple Lie groups in a product of harmonic maps with values in the components of the Iwasawa decomposition. In particular, we use this factorization to study the harmonic maps from $\mathbb{R}^n$ into $SL(2,\mathbb{R})$.
Our purpose is to use a Darboux homogenous derivative to understand the harmonic maps with values in homogeneous space. We present a characterization of these harmonic maps from the geometry of homogeneous space. Furthermore, our work covers all type of invariant geometry in homogeneous space.
We give a stochastic generalization of transport theorem on smooth manifold. Furthermore, we deduce a system of continuity equation and present some application on torus.
Let $G$ be a Lie Group with a left invariant connection $\nabla^{G}$. Denote by $\g$ the Lie algebra of $G$, which is equipped with a connection $\nabla^{\g}$. Our main is to introduce the concept of the Itô exponential and the Itô logarithm, which take in account the geometry of the Lie group $G$ and the Lie algebra $\g$. This definition characterize directly the martingales in $G$ with respect to the left invariant connection $\nabla^{G}$. Further, if any $\nabla^{\g}$ geodesic in $\g$ is send in a $\nabla^{G}$ geodesic we can show that the Itô exponential and the Itô logarithm are the same that the stochastic exponential and the stochastic logarithm due to M. Hakim-Dowek and D. Lépingle in [10]. Consequently, we have a Campbell-Hausdorf formula. From this formula we show that the set of affine maps from $(M,\nabla^{G})$ into $(G,\nabla^{G})$ is a subgroup of the Loop group. As in general, the Lie algebra is considered as smooth manifold with a flat connection, we show a Campbell-Hausdorf formula for a flat connection on $\g$ and a bi-invariant connection on $G$. To this main we introduce the definition of the null quadratic variation property. To end, we use the Campbell-Hausdorff formula to show that a product of harmonic maps with value in $G$ is a harmonic map.