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ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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The Jacobi Equation for Horizontal Geodesics on a Nonholonomic Distribution and the Schouten Curvature Tensor

Author(s):

Victor Revoltovich Krym

PhD in Physics and Mathematics
St.-Petersburg Mathematical Society
Fontanka, 27, 191023, St.-Petersburg

vkrym12@rambler.ru

Abstract:

The paper shows that if the distribution is defined on a manifold with the special smooth structure and does not depend on the vertical coordinates, then the Schouten curvature tensor coincides with the Riemannian curvature tensor. The Schouten curvature tensor is used to write the Jacobi equation for the distribution. This leads to studies on second-order optimality conditions for the horizontal geodesics in sub-Riemannian geometry. New example of a distribution with abnormal geodesics is constructed, their optimality being an open problem.

Keywords

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