U. Abresch and W. T. Meyer, Pinching below $\frac14$, injectivity radius, and conjugate radius, Journal of Differential Geometry, vol.40, issue.3, pp.643-691, 1994.
DOI : 10.4310/jdg/1214455781
URL : http://projecteuclid.org/download/pdf_1/euclid.jdg/1214455781

. Th and . Aubin, Nonlinear analysis on manifolds. Monge?Ampère equations, Grundlehren der math, Wissensch, vol.252, 1982.

R. L. Bishop, A relation between volume, mean curvature, and diameter, Amer, Math. Soc. Notices, vol.10, issue.364, 1963.

Y. Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs, C. R. Acad. Sci. Paris série I, vol.305, pp.805-808, 1987.

Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Communications on Pure and Applied Mathematics, vol.117, issue.4, pp.375-417, 1991.
DOI : 10.1002/cpa.3160440402

Y. Brenier and G. Loeper, A Geometric approximation to the euler equations: the Vlasov?Monge?Amp???re system, Geometrical and Functional Analysis GAFA, vol.14, issue.6, pp.1182-1218, 2004.
DOI : 10.1007/s00039-004-0488-1

L. Caffarelli, Some regularity properties of solutions of Monge Amp??re equation, Communications on Pure and Applied Mathematics, vol.131, issue.8-9, pp.965-969, 1991.
DOI : 10.1002/cpa.3160440809

L. Caffarelli, The regularity of mappings with a convex potential, Journal of the American Mathematical Society, vol.5, issue.1, pp.99-104, 1992.
DOI : 10.1090/S0894-0347-1992-1124980-8

J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry, Library, vol.9, 1975.
DOI : 10.1090/chel/365

D. Cordero-erausquin, Sur le transport de mesures p??riodiques, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.329, issue.3, pp.199-202, 1999.
DOI : 10.1016/S0764-4442(00)88593-6

D. Cordero-erausquin, R. Mccann, and M. Schmuckenschläger, A Riemannian interpolation inequalityàinequality`inequalityà la Borell, Brascamp and Lieb, Inventiones math, pp.219-257, 2001.

. Ph and . Delanoë, Gradient rearrangement for diffeomorphisms of a compact manifold, Diff. Geom. Appl, vol.20, issue.2, pp.145-165, 2004.

. Ph, . Delanoë, and . Doi, Lie solutions of Riemannian transport equations on compact manifolds, Diff, Geom. Appl

. Ph, G. Delanoë, and . Loeper, Gradient estimate for potentials of invertible gradient-mappings on the sphere, Calc. Var. PDE's 26, pp.297-311, 2006.

D. Ebin and J. Marsden, Groups of Diffeomorphisms and the Motion of an Incompressible Fluid, The Annals of Mathematics, vol.92, issue.1, pp.102-163, 1970.
DOI : 10.2307/1970699

W. Gangbo and R. Mccann, The geometry of optimal transportation, Acta Mathematica, vol.177, issue.2, pp.113-161, 1996.
DOI : 10.1007/BF02392620

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der math, Wissensch, vol.224, 1977.

K. Grove and K. Shiohama, A Generalized Sphere Theorem, The Annals of Mathematics, vol.106, issue.1, pp.201-211, 1977.
DOI : 10.2307/1971164

P. Günther, Einige Sätzë uber das Volumenelement eines Riemannschen Raumes, Pub. Math. Debrecen, vol.7, pp.78-93, 1960.

J. Jost, Riemannian geometry and geometric analysis, 1995.
DOI : 10.1007/978-3-642-21298-7

H. Young and . Kim, Counterexamples to continuity of optimal transportation on positively curved Riemannian manifolds, 2007.

H. Young, R. Kim, and . Mccann, Continuity, curvature and the general covariance of optimal transportation, Preprint, 2007.

W. Klingenberg, CONTRIBUTIONS TO RIEMANNIAN GEOMETRY IN THE LARGE, Ann. of Math, vol.69, pp.654-666, 1959.
DOI : 10.1142/9789812812797_0018

P. Delanoë and Y. Ge, Klingenberg, ¨ Uber Riemannsche Mannigfaltigkeiten mit positiver Krümmung, Comm. Math. helv, vol.35, pp.47-54, 1961.

G. Loeper, On the regularity of maps solutions of optimal transportation problems, 2006.

G. Loeper and C. Villani, Regularity of optimal transport in curved geometry: The nonfocal case, Duke Mathematical Journal, vol.151, issue.3, 2007.
DOI : 10.1215/00127094-2010-003

X. Ma, N. S. Trudinger, and X. 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

R. Mccann, Polar factorization of maps on Riemannian manifolds, Geometric and Functional Analysis, vol.11, issue.3, pp.589-608, 2001.
DOI : 10.1007/PL00001679

A. V. Pogorelov, The Minkowski Multidimensional Problem, Scripta Series in Math, 1978.

N. S. Trudinger, Recent developments in elliptic partial differential equations of Monge???Amp??re type, Proc. Intern. Congress Math. Math. Soc. Edit, pp.292-301, 2006.
DOI : 10.4171/022-3/15

A. D. Weinstein, The Cut Locus and Conjugate Locus of a Riemannian Manifold, The Annals of Mathematics, vol.87, issue.1, pp.1-29, 1968.
DOI : 10.2307/1970592