1 INRIA, 2004 route des lucioles, 06902 Sophia Antipolis, France
2 Institut de Mathématiques de Bourgogne, 9 avenue Savary, 21078 Dijon, France
Received: 28 February 2011
Revised: 24 November 2011
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
© EDP Sciences, SMAI, 2013