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 point out a few references.
It is resonable to look at :
The Mathematica GuideBooks by M.Trott, especially "Symbolics". One can find there some interesting exercises with complete solutions dealing with geometry. On the same internet site there is (also helpful in differential geometry) :
another interesting thing by the same author :
Obviously you could find many helpful things here at the Wolfram Demonstrations :
For example :
- Gauss Map and Curvature by Michael Rogers
- Complex Rotation of Minimal Surfaces by Roman Maeder
- Views of the Costa Minimal Surface by Enrique Zeleny
- Penrose Diagram by Christoph Meyer
Next it would be very helpful to look at this article in the Mathematica Journal by Murray Eisenberg and David J. M. Park, Jr. :
One should also look at the geometry section of
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 next version of Mathematica):