Metrics with equatorial singularities on the sphere

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 Poincare, Anal Non Lineaire 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.