Intuitively it might be clear that differential geometry is a very applicable subject in engineering and industry. I'd like to know how some industries/companies use differential geometry. I'd guess they'd be working on geometric modeling OR geometric data analysis. But I'd appreciate if someone can give some explicit examples, and also maybe point out some resources to learn more in that direction.

The only part I'm aware of is that differential geometry is used in medical imaging, computer vision, computer graphics research, where certain models use specific manifolds (shape manifolds for example). But other than that, I've really no idea how say the certain concepts like parallel transport or exponential maps get used in direct industrial applications. Also, if they are indeed used, what kind of Riemannian manifolds do they consider?

Any pointers and resources would be greatly appreciated!

WSEAS Int. Conf. on Dynamical systems and control, pp. 106-112. 2005. (PDF download link.) $\endgroup$ – Joseph O'Rourke Aug 4 '15 at 20:40