Shortest Paths for Sub-Riemannian Metrics on Rank-Two Distributions

Released on by Wensheng Liu
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  • Shortest Paths for Sub-Riemannian Metrics on Rank-Two Distributions Book Detail

  • Author : Wensheng Liu
  • Release Date : 1995
  • Publisher : American Mathematical Soc.
  • Genre : Mathematics
  • Pages : 121
  • ISBN 13 : 0821804049
  • File Size : 9,9 MB
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Shortest Paths for Sub-Riemannian Metrics on Rank-Two Distributions by Wensheng Liu PDF Summary

Book Description: A sub-Riemannian manifold ([italic capitals]M, E, G) consists of a finite-dimensional manifold [italic capital]M, a rank-two bracket generating distribution [italic capital]E on [italic capital]M, and a Riemannian metric [italic capital]G on [italic capital]E. All length-minimizing arcs on ([italic capitals]M, E, G) are either normal extremals or abnormal extremals. Normal extremals are locally optimal, i.e., every sufficiently short piece of such an extremal is a minimizer. The question whether every length-minimizer is a normal extremal was recently settled by R. G. Montgomery, who exhibited a counterexample. The present work proves that regular abnormal extremals are locally optimal, and, in the case that [italic capital]E satisfies a mild additional restriction, the abnormal minimizers are ubiquitous rather than exceptional. All the topics of this research report (historical notes, examples, abnormal extremals, Hamiltonians, nonholonomic distributions, sub-Riemannian distance, the relations between minimality and extremality, regular abnormal extremals, local optimality of regular abnormal extremals, etc.) are presented in a very clear and effective way.

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