Some early papers concerning SDEs on manifolds are:

• Itô, Kiyoshi (1950). Stochastic differential equations in a differentiable manifold. Nagoya Mathematical Journal. 1: 35–47.
• Itô, Kiyoshi (1962). The Brownian motion and tensor fields on Riemannian manifold. Proc. Int. Congr. Math., Stockholm 2.
• Molchanov S. A. (1975). Diffusion Processes and Riemannian Geometry. Russian Mathematical Surveys. Vol. 30, 1-63.
• Hsu E. P. (1987). Brownian motion and Riemannian geometry. 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, Cambridge University Press, Cambridge.
• M. Emery (1989). Stochastic calculus in manifolds, Springer, Berlin.
• E. P. Hsu (2002). Stochastic analysis on manifolds, 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. Advances in Modern Statistical Theory and Applications: A Festschrift in Honor of Morris L. Eaton. Institute of Mathematical Statistics. 102-125.

Some early papers concerning SDEs on manifolds are:

• Itô, Kiyoshi (1950). Stochastic differential equations in a differentiable manifold. Nagoya Mathematical Journal. 1: 35–47.
• Itô, Kiyoshi (1962). The Brownian motion and tensor fields on Riemannian manifold. Proc. Int. Congr. Math., Stockholm 2.

and see also,

• Molchanov S. A. (1975). Diffusion Processes and Riemannian Geometry. Russian Mathematical Surveys. Vol. 30, 1-63.
• Hsu E. P. (1987). Brownian motion and Riemannian geometry. 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, Cambridge University Press, Cambridge.
• M. Emery (1989). Stochastic calculus in manifolds, Springer, Berlin.
• E. P. Hsu (2002). Stochastic analysis on manifolds, 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. Advances in Modern Statistical Theory and Applications: A Festschrift in Honor of Morris L. Eaton. Institute of Mathematical Statistics. 102-125.

Some early papers concerning SDEs on manifolds are:

• Itô, Kiyoshi (1950). Stochastic differential equations in a differentiable manifold. Nagoya Mathematical Journal. 1: 35–47.
• Itô, Kiyoshi (1962). The Brownian motion and tensor fields on Riemannian manifold. Proc. Int. Congr. Math., Stockholm 2.
• Molchanov S. A. (1975). Diffusion Processes and Riemannian Geometry. Russian Mathematical Surveys. Vol. 30, 1-63.
• Hsu E. P. (1987). Brownian motion and Riemannian geometry. 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, Cambridge University Press, Cambridge.
• M. Emery (1989). Stochastic calculus in manifolds, Springer, Berlin.
• E. P. Hsu (2002). Stochastic analysis on manifolds, Graduate Studies in Mathematics. Providence, RI: American Mathematical Society, vol. 38.

Some early papers concerning SDEs on manifolds are:

• Itô, Kiyoshi (1950). Stochastic differential equations in a differentiable manifold. Nagoya Mathematical Journal. 1: 35–47.
• Itô, Kiyoshi (1962). The Brownian motion and tensor fields on Riemannian manifold. Proc. Int. Congr. Math., Stockholm 2.
• Molchanov S. A. (1975). Diffusion Processes and Riemannian Geometry. Russian Mathematical Surveys. Vol. 30, 1-63.
• Hsu E. P. (1987). Brownian motion and Riemannian geometry. 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, Cambridge University Press, Cambridge.
• M. Emery (1989). Stochastic calculus in manifolds, Springer, Berlin.
• E. P. Hsu (2002). Stochastic analysis on manifolds, 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. Advances in Modern Statistical Theory and Applications: A Festschrift in Honor of Morris L. Eaton. Institute of Mathematical Statistics. 102-125.