It's a frequent application since machines that scan the brain produce diffusion tensor images, i.e. I then googled computational riemannian geometry and came about this paper :, which uses differential geometry for computational anatomy. Although, the computation relies on the fact that the metric tensor function is simply a weighted average of metric tensors known at certain points in space, and the average is done with Gaussian basis functions, which ensures the smoothness of the metric. I liked this paper because it was the first one I saw of an actual numerical computation of geodesics, exponential and logarithmic maps. I've had a similar experience to yours, although I probably didn't go that deep into theory, but I recently read a paper that does some metric learning, with a differential geometry approach : Jefferey Lee - manifolds and differential geometry (up to chapter 8).Liviu - geometry of manifolds (roughly the first third of the book).Is it the case you need a lot of machinary before you can really tackle more examples?Įdit: Here are the main books I've studied from (I admit I wasn't completely thorough - however I'd rarely skip an exercise problem from a chapter i've been reading): Why is the pool of examples I found so far in textbooks so small? Most of the textbook problems I solved were general theory, which is great and rewarding, but I do feel unbalanced at the moment. And even with them i feel my experience is quite brief. The only examples I know and tampered with before are spheres and projective spaces (and a pinch of some matrix groups and grassmanians). I've had little to no experience with those and frankly they quite scare me. My problem lies with concrete examples and computations. proving Fundamental theorem of Riemannian geometry, proving general properties of connections and their curvature tensors, proving various identities about different derivations, proving frobenius theorem etc.). For example I can switch from global to local description with comfort, and do most symbolic calculations using the basic objects of the theory and not get confused about what i'm doing since i understand the operations and the context (e.g. I think I have a pretty solid grasp of the basic objects. Lately I started to realize how huge (and daunting) differential geometry really is. I studied a fair amount of the basic general theory and gone through a lot of the exercises from several textbooks. I've been studying differential geometry by myself for some time now.
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