Some early papers concerning **SDEs on manifolds** are: - Itô, Kiyoshi (1950). [Stochastic differential equations in a differentiable manifold.][1] Nagoya Mathematical Journal. 1: 35–47. - Itô, Kiyoshi (1962). [The Brownian motion and tensor fields on Riemannian manifold.][2] Proc. Int. Congr. Math., Stockholm 2. and see also, - Molchanov S. A. (1975). [Diffusion Processes and Riemannian Geometry.][3] Russian Mathematical Surveys. Vol. 30, 1-63. - Hsu E. P. (1987). [Brownian motion and Riemannian geometry.][4] in Geometry of Random Motion, edited by R. Durrett and M. Pinsky, Contemporary Math., American Mathematical Society, 73, 95--104. In addition, here are some highly cited books on **stochastic differential geometry** - Malliavin P. (1978). Géometrie différentielle stochastique. Presses de l'Université de Montréal, Montréal. - Elworthy K. D. (1982). [Stochastic differential equations on manifolds][5], Cambridge University Press, Cambridge. - M. Emery (1989). [Stochastic calculus in manifolds][6], Springer, Berlin. - E. P. Hsu (2002). [Stochastic analysis on manifolds][7], Graduate Studies in Mathematics. Providence, RI: American Mathematical Society, vol. 38. On the topic of **sampling on manifolds**, see the following expository paper and the references given in their discussion of related literature: - Diaconis, P., Holmes, S. and Shahshahani, M. (2013). [Sampling from a manifold.][8] Advances in Modern Statistical Theory and Applications: A Festschrift in Honor of Morris L. Eaton. Institute of Mathematical Statistics. 102-125. [1]: http://projecteuclid.org/euclid.nmj/1118764702 [2]: http://www.mathunion.org/ICM/ICM1962.1/Main/icm1962.1.0536.0539.ocr.pdf [3]: http://iopscience.iop.org/article/10.1070/RM1975v030n01ABEH001400/meta [4]: http://www.math.northwestern.edu/~ehsu/Brownian%20Motion%20and%20Riemannian%20Geometry.pdf [5]: http://link.springer.com/chapter/10.1007/978-1-4612-2224-8_10#page-1 [6]: http://www.springer.com/gp/book/9783540516644 [7]: http://bookstore.ams.org/gsm-38 [8]: http://projecteuclid.org/euclid.imsc/1379942050