I am studying BM on Riemannian manifolds and I am curious how this theory started. In the references below (esp. in Hsu's exposition), you will find many applications of that theory such as a probabilistic proof of the Atiyah-Singer index theorem.


I am also curious about the industrial applications (if any yet) given that Riemannian geometry is used in Machine learning and statistics.


Thanks anyhow


In case you are interested
[1]<https://www.math.kyoto-u.ac.jp/probability/sympo/PSS03abstract.pdf>
[2]<http://www.math.northwestern.edu/~ehsu/Brownian%20Motion%20and%20Riemannian%20Geometry.pdf> (proof of Atiyah Singer index thm)



Origins and Industrial Applications of stochastic processes (eg. Brownian motion) on Riemannian manifolds
The earliest "industrial" application I know is in the context of microwave engineering: the eigenvalues of the transmission matrix through a waveguide with random scatterers perform a Brownian motion in hyperbolic space as a function of the length of the waveguide.


[Waveguides with Random Inhomogeneities and Brownian Motion In the Lobachevsky Plane](http://epubs.siam.org/doi/abs/10.1137/1104038) (1959).


