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.
Atlas 2 for Mathematica is the add-on for doing modern differential geometry calculations.
Calculations are coordinate free
Graphical user interfaces
Video and Screencasts
The modern differential geometry is a vast subject and while not specified exactly what you need the question is a bit too general. I would rather point out a few references.
If you are looking for a package for tensor calculus, especially in General Relativity, the best choice is xAct made by José M. Martín-García (as far as I know he actually develops built-in functionality for the future versions of Mathematica):
However if you need something straightforward and concise, look at this implementation below of a few fundamental objects in differential geometry. You need a metric
The above is a Lorentzian metric tensor (in a given map) of a static spherically symmetric four dimentional manifold, and the following are the inverse metric, Christoffel symbol of the second kind, Riemann and Ricci curvature tensors and the Ricci scalar with brief descriptions of their usage:
This is not an optimal implementation, but it is a good point to start building your own package. You could think about introducing a functional definition of the covariant derivative as well as lower and upper indices of covariant and contravariant tensors. One of the common difficulties with this is the multiplicity of definitions and conventions for Riemann and Ricci tensors etc, and that's why I added descriptions of the given functions.
Besides of the above I recommend to look at the Wolfram Demonstrations :
One should also look at the geometry section of
Next it would be even more helpful and strongly recommended to look at these articles (some of them are particularly devoted to differential geometry topics, others only deal with useful geometric techniques) in the Mathematica Journal :
You may want to look, also, at David Park's