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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)

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  • $\begingroup$ There seem to be two different questions you are asking here. One is "how this theory started"; the other is "are there industrial applications". Which one would you like answers to? $\endgroup$
    – Yemon Choi
    Commented Feb 10, 2015 at 20:13
  • $\begingroup$ yeah sorry, I was just fixing that. Thanks anyhow. $\endgroup$
    – Thomas Kojar
    Commented Feb 10, 2015 at 20:15

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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 (1959).

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A somewhat recent engineering application is in study of dynamics and control on manifolds such as SO(3), which is the manifold on which the (position) state of a rotating rigid body (such as a satellite) evolves. For inference/filtering problems (i.e. trying to estimate the position on SO(3) given some noisy measurements), it is required to perform uncertainty propagation on these manifolds. Here's a reference: http://arxiv.org/abs/0803.1515

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