| Issue |
ESAIM: COCV
Volume 19, Number 2, April-June 2013
|
|
|---|---|---|
| Page(s) | 533 - 554 | |
| DOI | https://doi.org/10.1051/cocv/2012020 | |
| Published online | 21 February 2013 | |
Conjugate-cut loci and injectivity domains on two-spheres of revolution∗,∗∗,∗∗∗
1 INRIA, 2004 route des lucioles, 06902 Sophia Antipolis, France
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2 Institut de Mathématiques de Bourgogne, 9 avenue Savary, 21078 Dijon, France
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Received: 28 February 2011
Revised: 24 November 2011
Abstract
In a recent article [B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26 (2009) 1081–1098], we relate the computation of the conjugate and cut loci of a family of metrics on two-spheres of revolution whose polar form is g = dϕ2 + m(ϕ)dθ2 to the period mapping of the ϕ-variable. One purpose of this article is to use this relation to evaluate the cut and conjugate loci for a family of metrics arising as a deformation of the round sphere and to determine the convexity properties of the injectivity domains of such metrics. These properties have applications in optimal control of space and quantum mechanics, and in optimal transport.
Mathematics Subject Classification: 58B20 / 49K15 / 53C22
Key words: Conjugate and cut loci / injectivity domain / optimal control / optimal transport
Work supported by ANR Geometric Control Methods (project No. NT09504490).
Also supported by Conseil Régional de Bourgogne (contract No. 2009-160E-160-CE-160T) and SADCO Initial Training Network (FP7 grant No. 264735-SADCO).
Also supported by a DGA grant.
© EDP Sciences, SMAI, 2013
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