P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control, Progress in Nonlinear Differential Equations and their Applications, 2004.

M. Castelpietra and L. Rifford, Regularity properties of the distance functions to conjugate and cut loci for viscosity solutions of Hamilton-Jacobi equations and applications in Riemannian geometry, ESAIM: Control, Optimisation and Calculus of Variations, vol.16, issue.3
DOI : 10.1051/cocv/2009020

URL : https://hal.archives-ouvertes.fr/hal-00923303

A. Figalli, Regularity of optimal transport maps (after Ma-Trudinger-Wang and Loeper)

A. Figalli, L. Rifford, and C. Villani, Nearly Round Spheres Look Convex, American Journal of Mathematics, vol.134, issue.1, 2009.
DOI : 10.1353/ajm.2012.0000

URL : https://hal.archives-ouvertes.fr/hal-00923321

A. Figalli, L. Rifford, and C. Villani, On the Ma???Trudinger???Wang curvature on surfaces, Calculus of Variations and Partial Differential Equations, vol.255, issue.9
DOI : 10.1007/s00526-010-0311-9

S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry, second ed. Universitext, 1990.

J. J. Hebda, Metric structure of cut loci in surfaces and Ambrose's problem, Journal of Differential Geometry, vol.40, issue.3, pp.621-642, 2005.
DOI : 10.4310/jdg/1214455780

J. Itoh and K. Kiyohara, The cut loci and the conjugate loci on ellipsoids, manuscripta mathematica, vol.114, issue.2, pp.247-264, 2004.
DOI : 10.1007/s00229-004-0455-z

J. Itoh and M. Tanaka, The Lipschitz continuity of the distance function to the cut locus, Transactions of the American Mathematical Society, vol.353, issue.01, pp.21-40, 2001.
DOI : 10.1090/S0002-9947-00-02564-2

Y. Li and L. Nirenberg, The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations, Communications on Pure and Applied Mathematics, vol.15, issue.1, pp.85-146, 2005.
DOI : 10.1002/cpa.20051

G. Loeper and C. Villani, Regularity of optimal transport in curved geometry: The nonfocal case, Duke Mathematical Journal, vol.151, issue.3
DOI : 10.1215/00127094-2010-003

X. N. Ma, N. S. Trudinger, and X. J. Wang, Regularity of Potential Functions of the Optimal Transportation Problem, Archive for Rational Mechanics and Analysis, vol.13, issue.2, pp.151-183, 2005.
DOI : 10.1007/s00205-005-0362-9

H. Poincaré, Sur Les Lignes Geodesiques Des Surfaces Convexes, Transactions of the American Mathematical Society, vol.6, issue.3, pp.237-274, 1905.
DOI : 10.2307/1986219

C. Villani, Optimal transport, old and new, Grundlehren des mathematischen Wissenschaften, vol.338, 2009.