What is the stress-energy-momentum tensor and its role in general relativity?

[kevinferreira]

http://en.wikipedia.org/wiki/Tangent_space Both definitions here (first and second) ascribe an algebraic (i.e. according to the axioms here http://en.wikipedia.org/wiki/Vector_space) character to the tangent space in a point x of a general manifold. Where does the norm come from which would induce the topology on T_x (M) ?

Your first sentence induces the set inclusion

{vector spaces} $\subset$ {topological spaces}

which is not not correct (the axioms of a vector space don't mention a norm, so there wouldn't be any norm-induced topology
).

The right set connection is:

{topological vector spaces} = {vector spaces} $\cap$ {topological spaces}

Oh, yes, you're absolutely right, I was wrong in admitting a norm as an axiom (or a direct consequence of an axiom) of vector space definition. I guess it comes from my time spent studying functional analysis!
Anyway, thanks for clearing that up for me.
But that's troubling, as in the wikipedia article of the exponential map it is said explicitly

The radius of the largest ball about the origin in TpM that can be mapped diffeomorphically...

So a notion of distance is used... I wonder how.

 

[micromass]

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