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.