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.
 - 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 (1987),
   95--104.

In addition, here are several 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][7], Springer,
   Berlin.
 - E. P. Hsu (2002). [Stochastic analysis on manifolds][8], Graduate
   Studies in Mathematics. Providence, RI: American Mathematical
   Society, vol. 38.

  [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://link.springer.com/chapter/10.1007/BFb0099433#page-1
  [7]: http://www.springer.com/gp/book/9783540516644
  [8]: http://bookstore.ams.org/gsm-38