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Conjugate-cut loci and injectivity domains on two-spheres of revolution∗∗∗∗∗∗

Published online by Cambridge University Press:  21 February 2013

Bernard Bonnard ,
Jean-Baptiste Caillau  and
Gabriel Janin
Bernard Bonnard
Affiliation:
INRIA, 2004 route des lucioles, 06902 Sophia Antipolis, France. bernard.bonnard@u-bourgogne.fr Institut de Mathématiques de Bourgogne, 9 avenue Savary, 21078 Dijon, France; jean-baptiste.caillau@u-bourgogne.fr; gabriel.janin@u-bourgogne.fr
Jean-Baptiste Caillau
Affiliation:
Institut de Mathématiques de Bourgogne, 9 avenue Savary, 21078 Dijon, France; jean-baptiste.caillau@u-bourgogne.fr; gabriel.janin@u-bourgogne.fr
Gabriel Janin
Affiliation:
Institut de Mathématiques de Bourgogne, 9 avenue Savary, 21078 Dijon, France; jean-baptiste.caillau@u-bourgogne.fr; gabriel.janin@u-bourgogne.fr
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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.

Keywords

Conjugate and cut lociinjectivity domainoptimal controloptimal transport
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 2 , April 2013 , pp. 533 - 554
Copyright
© EDP Sciences, SMAI, 2013

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