image processing - 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!

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Manipulate[ SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, Mesh -> None, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], {n, 0, 1}] I cant' seem to be able to make a filling between two spheres. I've already tried the obvious Filling -> {1 -> {2}} but Mathematica doesn't seem to like that option. Is there any easy way around this or ... Answer There is no built-in filling in SphericalPlot3D . One option is to use ParametricPlot3D to draw the surfaces between the two shells: Manipulate[ Show[SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], ParametricPlot3D[{ r {Sin[t] Cos[1.5 Pi], Sin[t] Sin[1.5 Pi], Cos[t]}, r {Sin[t] Cos[0 Pi], Sin[t] Sin[0 Pi], Cos[t]}}, {r, 1, 2 - n}, {t, 0, Pi}, PlotStyle -> Yellow, Mesh -> {2, 15}]], {n, 0, 1}]

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