Du, Shi-ZhongLi, Qi-Rui2015-06-172015-06-170944-2669http://hdl.handle.net/1885/13978In this paper we consider the optimal transportation on Riemannian surfaces when the cost function is squared distance. The main ingredient is the verification of MTW condition. It is known that MTW condition holds if Gauss curvature is sufficiently close to 1 in C² norm. In this paper we give an explicit condition on Gauss curvature such that MTW condition is satisfiedThis paper was supported by the Australian Research Council. The first author was partially supported by the National Natural Science Foundation of China (11101106). The second author was also supported by the Chinese Scholarship Council. The authors would like to show their appreciation to Professor Xu-Jia Wang for his encouragement and instructive discussions.© Springer-Verlag Berlin Heidelberg 20132013-11-1610.1007/s00526-013-0684-72015-12-08