One-parameter family of Clairaut-Liouville metrics

Bernard Bonnard (1) , Jean-Baptiste Caillau (2) , Minoru Tanaka (3)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 IRIT-APO - Algorithmes Parallèles et Optimisation
3 Tokai University

Riemannian metrics with singularities are considered on the $2$-sphere of revolution. The analysis of such singularities is motivated by examples stemming from mechanics and related to projections of higher dimensional (regular) sub-Riemannian distributions. An unfolding of the metrics in the form of an homotopy from the canonical metric on $\SS^2$ is defined which allows to analyze the singular case as a limit of standard Riemannian ones. A bifurcation of the conjugate locus for points on the singularity is finally exhibited.

Mots clés: $2$-sphere of revolution; unfolding; space mechanics

Domaines: Optimisation et contrôle [math.OC]

Bernard Bonnard, Jean-Baptiste Caillau, Minoru Tanaka. One-parameter family of Clairaut-Liouville metrics. 2007. ⟨hal-00177686v3⟩