Seminar talk, 2 November 2016: Difference between revisions

Latest revision as of 13:18, 25 October 2016

Title: Sub-Riemannian geometry: minimizers, spheres, cut loci

Abstract:
A sub-Riemannian structure on a smooth manifold $M$ is a vector distribution $\Delta \subset TM$ together with a metric $g$ on the distribution $\Delta$ .

Horizontal curves are Lipschitzian curves on $M$ tangent to the distribution $\Delta$ almost everywhere. If $M$ is connected and the Lie algebra generated by the distribution $\Delta$ spans the whole tangent space $TM$ , then every two points of $M$ can be connected by a horizontal curve (Rashevsky-Chow theorem).

The length of horizontal curve is the integral of the length of velocity vector. The sub-Riemannian distance (the Carnot-Carathéodory distance) $d(q_{0},q_{1})$ between points $q_{0},q_{1}\in M$ is said to be the lower bound of all lengths of horizontal curves joining $q_{0}$ with $q_{1}$ . A minimizer is a horizontal curve whose length equals to the distance between its ends. Rather weak conditions guarantee the existence of a minimizer between close enough points (Filippov theorem). If sub-Riemannian balls are compact then under conditions of Rashevsky-Chow theorem every points can be join with a minimizer.

Geodesic is a horizontal curve whose small arcs are minimizers. Geodesics are projections $T^{*}M\to M$ of trajectories of some natural Hamiltonian system on $T^{*}M$ (Pontryagin maximum principle).

The talk will discuss the following questions:

left invariant sub-Riemannian structures on Lie groups,
symmetry method for searching minimizers,
examples of studied sub-Riemannian geometries (three-dimensional Lie groups, Engel group, Cartan group),
restrictions of known methods (the Liouville non-integrability of flat sub-Riemannian structures of depth more than 3),
applications to mechanics, robototechnics, image processing.

@@ Line 4: / Line 4: @@
 | abstract = A sub-Riemannian structure on a smooth manifold <math>M</math> is a vector distribution <math>\Delta \subset TM</math> together with a metric <math>g</math> on the distribution <math>\Delta</math>.
-Horizontal curves are Lipschitzian curves on <math>M</math> tangent to the distribution <math>\Delta</math> almost everywhere.  If <math>M</math> is connected and the Lie algebra generated by the distribution <math>\Delta</math> span the whole tangent space <math>TM</math>, then every two points of <math>M</math> can be connected by a horizontal curve (Rashevsky-Chow theorem).
+Horizontal curves are Lipschitzian curves on <math>M</math> tangent to the distribution <math>\Delta</math> almost everywhere.  If <math>M</math> is connected and the Lie algebra generated by the distribution <math>\Delta</math> spans the whole tangent space <math>TM</math>, then every two points of <math>M</math> can be connected by a horizontal curve (Rashevsky-Chow theorem).
 The length of horizontal curve is the integral of the length of velocity vector.  The sub-Riemannian distance (the Carnot-Carathéodory distance) <math>d(q_0, q_1)</math> between points <math>q_0, q_1 \in M</math> is said to be the lower bound of all lengths of horizontal curves joining <math>q_0</math> with <math>q_1</math>.  A minimizer is a horizontal curve whose length equals to the distance between its ends.  Rather weak conditions guarantee the existence of a minimizer between close enough points (Filippov theorem).  If sub-Riemannian balls are compact then under conditions of Rashevsky-Chow theorem every points can be join with a minimizer.
-Geodesic is a horizontal curve whose small arcs are minimizers.  Geodesics are projections <math>T^*M \to M</math> of trajectories of some natural Hamiltonian system on <math>$T^*M</math> (Pontryagin maximum principle).
+Geodesic is a horizontal curve whose small arcs are minimizers.  Geodesics are projections <math>T^*M \to M</math> of trajectories of some natural Hamiltonian system on <math>T^*M</math> (Pontryagin maximum principle).
 The talk will discuss the following questions:

Seminar talk, 2 November 2016: Difference between revisions

Latest revision as of 13:18, 25 October 2016

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