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Annali di Matematica Pura ed Applicata (1923 -)

, Volume 193, Issue 5, pp 1353–1382 | Cite as

Metrics with equatorial singularities on the sphere

  • B. Bonnard
  • J.-B. CaillauEmail author
Article

Abstract

Motivated by optimal control of affine systems stemming from mechanics, metrics on the two-sphere of revolution are considered; these metrics are Riemannian on each open hemisphere, whereas one term of the corresponding tensor becomes infinite on the equator. Length-minimizing curves are computed, and structure results on the cut and conjugate loci are given, extending those in Bonnard et al. (Ann Inst H Poincaré Anal Non Linéaire 26(4):1081–1098, 2009). These results rely on monotonicity and convexity properties of the quasi-period of the geodesics; such properties are studied on an example with elliptic transcendency. A suitable deformation of the round sphere allows to reinterpretate the equatorial singularity in terms of concentration of curvature and collapsing of the sphere onto a two-dimensional billiard.

Keywords

Two-sphere of revolution Almost- and sub-Riemannian metrics  Cut and conjugate locus 

Mathematics Subject Classification (2000)

53C17 49K15 

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Mathematical Institute, University of Bourgogne, CNRSDijonFrance

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