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I've been studying differential geometry by myself for some time now. I studied a fair amount of the basic general theory and gone through a lot of the exercises from several textbooks.

Lately I started to realize how huge (and daunting) differential geometry really is. I think I have a pretty solid grasp of the basic objects. 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. 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.).

My problem lies with concrete examples and computations. I've had little to no experience with those and frankly they quite scare me. The only examples I know and tampered with before are spheres and projective spaces (and a pinch of some matrix groups and grassmanians). And even with them i feel my experience is quite brief. Most of the textbook problems I solved were general theory, which is great and rewarding, but I do feel unbalanced at the moment.

Why is the pool of examples I found so far in textbooks so small?

Is it the case you need a lot of machinary before you can really tackle more examples?

What would you recommend me to do?

Edit: 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):

  • guilliam and pollack
  • Liviu - geometry of manifolds (roughly the first third of the book).
  • Jefferey Lee - manifolds and differential geometry (up to chapter 8).
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  • $\begingroup$ It would be helpful to say precisely which textbooks you're referring to. $\endgroup$ – user98602 Sep 29 '15 at 15:28
  • $\begingroup$ Even parametric surfaces embedded in $\mathbb R^3$ are hard to compute explicitly because the expression of the normal vector is not nice due to normalization. $\endgroup$ – lhf Sep 29 '15 at 17:17
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    $\begingroup$ One issues is the sheer number of degrees of freedom involved. For example, the Riemann curvature tensor on a generic $n$-manifold consists of something on the order of $n^4/12$ independent functions. This is too unwieldy to use in practice unless $n$ is small. But, if you know your manifold is highly symmetric, this bound can reduce significantly, leading to a tractable problem. $\endgroup$ – Jason DeVito Sep 29 '15 at 17:30
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    $\begingroup$ I just want to add: I think this question really hits at a problem in the pedagogy of differential geometry. I think it is highly typical that people know the general theory well, but can't do simple concrete computations. I recently asked a friend of mine, who just passed his PhD qualifiers in DG at a prestigious school, to tell me what the covariant derivative on the sphere (in the standard embedding) is. It took him quite a long time to figure out the answer (and his answer was off by a sign). If you Google "covariant derivative on the sphere" you, similarly, find nothing. $\endgroup$ – James Fennell Mar 19 '18 at 14:58
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    $\begingroup$ I think this is a problem because (a) it makes DG inaccessible when in reality a lot of the concepts are not too hard and (b) a lot of research, for example in geometric PDE, requires basic DG knowledge like "what is the covariant derivative on the sphere" but the current crop of books and web resources aren't designed to answer theses kinds of questions. $\endgroup$ – James Fennell Mar 19 '18 at 15:00
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The book Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers seems perfectly suited for your purposes. It has chapters with many concrete examples and computations on

  1. Differentiable Manifolds
  2. Tensor Fields and Differential Forms
  3. Integration on Manifolds
  4. Lie Groups
  5. Fibre Bundles
  6. Riemannian Geometry
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You may try Reyer Sjamaar's note "Manifolds and Differential Forms":

http://www.math.cornell.edu/~sjamaar/manifolds/manifold.pdf


Add: A recent good book is Differential Forms: Theory and Practice (2014) by Steven H. Weintraub. I come across this book from a book review in American Mathematical Monthly:

Garrity, Thomas. Differential Forms: Theory and Practice. American Mathematical Monthly, Volume 123, Number 4, April 2016, pp. 407-412(6).

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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 : http://files.is.tue.mpg.de/shauberg/papers/NIPS2012/paper.pdf

I liked this paper because it was the first one I saw of an actual numerical computation of geodesics, exponential and logarithmic maps. 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 then googled computational riemannian geometry and came about this paper : https://hal.inria.fr/inria-00616104/document, which uses differential geometry for computational anatomy. It's a frequent application since machines that scan the brain produce diffusion tensor images, i.e. one tensor per sampled point.

I don't know if it answers your question, but it seems to me that you were looking for numerical applications of differential geometry, I hope that answers it.

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