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### Differential geometry add-ons for Mathematica

I need a differential geometry tool for Mathematica. Is there any add-on? Unfortunately, Mathematica does not include such functionality or I can not find it.
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### Computing Gaussian curvature

Can Gaussian curvature $K$ be computed from WolframAlpha or any other available Mathematica program? Please indicate the program or its reference. If input ...
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### Using Mathematica for tensor analysis

Has anybody used tensors in Mathematica? How to properly work with them? I find Mathematica not very friendly in this field, as I am defining my own functions for lowering & raising indices, ...
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### The time-like geodesics (orbits) in the Schwarzschild spacetime

I am trying to plot Schwarzschild's orbit without invoking the geodesic equation. As a reference I am using Chandrasekhar's Book (The Mathematical theory of Black Holes, Oxford University Press). In ...
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### Contracting with Levi-Civita (totally antisymmetric) tensor

I have an array $v_{ijk}$ which is effectively a rank-$3$ tensor with dimensions $3\times3\times3$, and I need to contract it with $e^{ijk}$, i.e. evaluate $v_{ijk} e^{ijk}$ (see Einstein convention)....