“…Also convexity property of the injectivity domain of the exponential map is related to the continuity property of the Monge transport map T on the surfaces [6]. The structure of the conjugate and cut loci surfaces diffeomorphic to S 2 was investigated in details Bernard Bonnard Institut de mathématiques de Bourgogne, 9 avenue Savary, 21078 Dijon, France, e-mail: bernard.bonnard@u-bourgogne.fr Olivier Cots INRIA, 2004 route des lucioles, F-06902 Sophia Antipolis, France, e-mail: olivier.cots@inria.fr Lionel Jassionnesse Institut de mathématiques de Bourgogne, 9 avenue Savary, 21078 Dijon, France, e-mail: lionel.jassionnesse@u-bourgogne.fr 1 by Poincaré and Myers [9], [10]. In the analytic case, the cut locus is a finite tree and the extremity of each branch is a cusp point.…”