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An Inverse Problem in the Calculus of Variations and the Characteristic Curves of Connections on SO(3)-Bundles

Published online by Cambridge University Press:  20 November 2018

Richard Atkins  and
Zhong Ge
Richard Atkins
Affiliation:
The Fields Institute for Research in Mathematical Sciences, 185 Columbia St. West, Waterloo, Ontario N2L 5Z5
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Abstract

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This paper concerns an inverse problem in the calculus of variations, namely, when a two-dimensional symmetric connection is globally a Riemannian or pseudo-Riemannian connection. Two new local characterizations of such connections in terms of the Ricci tensor and the Riemann curvature tensor respectively are given, together with a solution to the global problem. As an application, the problem of whether the characteristic curves of a connection on an SO(3)-bundle on a surface are the geodesies of a Riemannian metric on the surface is studied. Some applications to non-holonomic dynamics are discussed.

Keywords

53C0553C22
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 38 , Issue 2 , 01 June 1995 , pp. 129 - 140
Copyright
Copyright © Canadian Mathematical Society 1995

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