I've no knowledge in mathematica (but I do in matlab), but I'd really appreciate if someone could mention what is/are the best and easy to learn mathematica package(s) for symbolic and numerical (both, really) computation of Riemannian geometry, specially Christoffel symbols, sectional curvature, and parallel transport along a given curve on M, given the topological type of the manifold M and the Riemannian metric g on M.

To explain myself a little more: in order to symbolically compute the Christoffel symbols, I've to invert a matrix and compute the symbolic and numerical derivatives w.r.t. the matrix. These matrices come from observations of medical data and are d by n matrices with n being a huge number, and d is normally 2 or 3.

After that, I've to compute the parallel transport along a curve c, which'll involve solving a system of first order linear ordinary differential equation with matrix entries depending on the derivative c' and the Christoffel symbols.

Thank you!

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    $\begingroup$ Maybe 8895 will help. $\endgroup$ – b.gates.you.know.what Apr 25 '15 at 13:43
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    $\begingroup$ Welcome to Mathematica.SE! I suggest the following: 1) As you receive help, try to give it too, by answering questions in your area of expertise. 2) Read the faq! 3) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign! $\endgroup$ – user9660 Apr 25 '15 at 13:44
  • $\begingroup$ Will do, absolutely! $\endgroup$ – Mathmath Apr 25 '15 at 13:45
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    $\begingroup$ If you were a physicist specializing in general relativity, I would suggest xAct with xCoba for the Christoffels, but it requires extensive knowledge of differential geometry. There exist less complicated packages, but I have no experience with them. $\endgroup$ – auxsvr Apr 25 '15 at 22:09
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    $\begingroup$ Please read my answer below. I am not familiar with using diff. Geometry with medical data. Would you perhaps briefly explain what these large data matrices represent and what kind of manifolds you want to detect/describe with Christoffels/ curvature $\endgroup$ – magma Apr 30 '15 at 7:18

The most powerful tensor package suit for MMA (and arguably for any CAS) is xAct. It uses the full machinery of diff geometry (fiber bundles, connections, forms, ect) and a powerful canonicalization algorithm, both symbolically and numerically. Obviouly you can use just a fraction of this power. I am the developer of xPrint, the GUI interface to xAct. With xPrint you can use many of xAct tools with "point and click" ease. It would shield you from some MMA technicalities. We have an active forum to help newcomers. The packages are free.

xAct.es is the main site

xPrint main site

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    $\begingroup$ meta.mathematica.stackexchange.com/questions/428/… is a list to most popular packages for MMA $\endgroup$ – magma Apr 30 '15 at 8:15
  • $\begingroup$ Thank you, will surely try it. Although it's free, I've to still install mathematica (not free), right? $\endgroup$ – Mathmath May 1 '15 at 1:42
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    $\begingroup$ Yes you need to install mathematica. It is not free, but there are several licence options. All equally powerful...very powerful. $\endgroup$ – magma May 1 '15 at 3:47

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