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the (free) software package, grtensor

GRTensor is a software package that "sits on top of" Maple. It is the best package I know to calculate connections and curvatures from metrics. It is free software that may be downloaded directly from this link. Go there and follow the links, looking around. The software is downloaded from the link called Software in the frame on the left-hand side.
It runs as a special package inside any version of Maple equal or higher than Release 6. They say that they have a version that also runs inside Mathematica, but I haven't tried it. The directions are fairly clear, I believe, but ask me if you cannot get it to run correctly. Basically you must copy the file "maple.ini" to the subdirectory called Users inside your directory of Maple.

I have also put online a sample grtensor session, where I use it to calculate the connections and curvatures for the general static, spherically-symmetric metric.

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