# Package for symbolic computation of christoffel symbols and parallel transports in Riemannian geometry, given the metric

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!

• Maybe 8895 will help. – b.gates.you.know.what Apr 25 '15 at 13:43
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• Will do, absolutely! – Mathmath Apr 25 '15 at 13:45
• 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. – auxsvr Apr 25 '15 at 22:09
• 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 – magma Apr 30 '15 at 7:18