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GEOMETRIC INTERPRETATIONS OF THE SYMMETRIC PRODUCT IN AFFINE DIFFERENTIAL GEOMETRY AND APPLICATIONS

MARÍA BARBERO-LIÑÁN

Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. Universidad 30, 28911 Leganés, Spain

Institute for the Mathematical Sciences (CSIC-UAM-UC3M-UCM), C/Nicolás Cabrera 13-15, 28049 Madrid, Spain

Work performed while a postdoctoral fellow at Queen's University.

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and

ANDREW D. LEWIS

Department of Mathematics and Statistics, Queen's University, Kingston, ON K7L 3N6, Canada

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https://doi.org/10.1142/S0219887812500739Cited by:5 (Source: Crossref)

Abstract

The symmetric product of vector fields on a manifold arises when one studies the controllability of certain classes of mechanical control systems. A novel geometric description of the symmetric product is provided using parallel transport, along the lines of the flow interpretation of the Lie bracket. This geometric interpretation of the symmetric product yields two different applications. First, an intrinsic proof is provided of the fact that the distributions closed under the symmetric product are exactly those distributions invariant under the geodesic flow. Second, some applications in geometric control theory for mechanical systems are clarified.

Keywords:

AMSC: 53B05, 53C22

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