Where do I start with Non-Euclidean Geometry? I've been trying to grok General Relativity for a while now, and I've been having some trouble. Many physics textbooks gloss over the subject with an "it's too advanced for this medium", and many other resources start out with something like "well space is just a non-euclidean manifold with a Ricci tensor defined as follows:", which would be cool if I understood non-Euclidean manifolds or Ricci tensors.
Unfortunately, when I try to crack open Wikipedia articles on non-Euclidean articles on the Ricci tensor I have trouble making sense of all the foreign terms.
Is non-euclidean geometry a good place to start if I want to understand general relativity? What's a good introductory resource for non-Euclidean geometry for someone who's only ever dealt with Euclidean?
(Note: I understand the basic principles of general relativity, i.e. how acceleration and gravity are different perspectives about the same thing and how clocks move slower when higher in a gravity field, but I want to understand the math and how it was derived.)
 A: I wouldn't start with learning the maths. Mathematicians take a very different approach from physicists and I doubt it would help much.
You just need the right textbook. I strongly recommend "A first course in general relativity" by Bernard F. Schutz. This seems to me to strike the right balance between understanding the physics and understanding the maths. Note however that even a "first course" is pretty hard work - to get through the book will require much sweat, and I speak from experience :-)
A: It really depends on your background in mathematics, what references are appropriate. In my oppinion you need at least a solid understanding of linear algebra and vector calculus, as most books on general relativity will assume.
A very nice although quite long book on general relativity is "Gravitation" by Misner, Thorne and Wheeler. It is very pedagocical and has intuitive descriptions of all geometric concepts you need to understand. In this regard it is better than many books on differential geometry. The only drawback is that a proper development of those foundations takes a lot of time and they take quite a few detours, so it is probably not suitable for someone, who is impatient to learn about general relativity.
On the other hand the less thourough books usually reduce the mathematical part to cooking recipes and give you little to no idea, why they actually work. If you have had no formal education in mathematics on university level, I suspect it will be hard to understand general relativity on more than a superficial level. As a general rule it is usually not a good idea to rush to the subjects like general relativity immediatly. It makes understanding them unneccessarily hard, compared to an approach where one tries to understand the fundamentals first. 
A: I recomended to study non-euclidan geometry from the book "Non-Euclidian Geometry" by
 Stefan Kulczycky (Transleted from Polish) - Pergamon Press     Haim Reouven-Israel 
