arXiv subjects
Publications and source records attributed to Lenka Zalabova.
We study the role of symmetries in control systems through the geometric algebra approach. We discuss two specific control problems on Carnot groups of step $2$ invariant with respect to the action of $SO(3)$. We understand the geodesics as the curves in suitable geometric algebras which allows us to assess a new algorithm for the local control.
We use the methods of geometric control theory to study extremal trajectories of vertical rolling disk. We focus on the role of symmetries of the underlying geometric structures. We demonstrate the computations in the CAS Maple package DifferentialGeometry.
We study local control of the mechanism with the growth vector (4,7). We study controllability and extremal trajectories on the nilpotent approximation as an example of the control theory on Lie group. We give solutions of the system an show examples of local extremal trajectories.
The symmetry dimension of a geometric structure is the dimension of its symmetry algebra. We investigate symmetries of almost quaternionic structures of quaternionic dimension $n$. The maximal possible symmetry is realized by the quaternionic projective space $\mathbb{H}P^n$, which is flat and has the symmetry algebra $\mathfrak{sl}(n+1,\mathbb{H})$ of dimension $4n^2+8n+3$. For non-flat almost quaternionic manifolds we compute the next biggest (submaximal) symmetry dimension. We show that it is equal to $4n^2-4n+9$ for $n>1$ (it is equal to 8 for $n=1$). This is realized both by a quaternionic structure (torsion--free) and by an almost quaternionic structure with vanishing quaternionic Weyl curvature.
We study here systems of symmetries on $|1|$--graded parabolic geometries. We are interested in smooth systems of symmetries and we discuss non--flat homogeneous $|1|$--graded geometries. We show the existence of an invariant admissible affine connection under quite weak condition on the system.
The classical concept of affine locally symmetric spaces allows a generalization for various geometric structures on a smooth manifold. We remind the notion of symmetry for parabolic geometries and we summarize the known facts for $|1|$--graded parabolic geometries and for almost Grassmannian structures, in particular. As an application of two general constructions with parabolic geometries, we present an example of non--flat Grassmannian symmetric space. Next we observe there is a distinguished torsion--free affine connection preserving the Grassmannian structure so that, with respect to this connection, the Grassmannian symmetric space is an affine symmetric space in the classical sense.
We generalize the concept of affine locally symmetric spaces for parabolic geometries. We discuss mainly $|1|$--graded geometries and we show some restrictions on their curvature coming from the existence of symmetries. We use the theory of Weyl structures to discuss more interesting $|1|$--graded geometries which can carry a symmetry in a point with nonzero curvature. More concretely, we discuss the number of different symmetries which can exist at the point with nonzero curvature.