I know that the metric is an intrinsic property of a manifold, hence it should be possible to compute it without any use of an embedding in a higher dimensional space.

That is, one can easily compute the metric on a surface of a 2-sphere just by computing the inner products of the basis tangent vectors as they appear in R^3, with the trivial Euclidean inner product.

My question is, can one compute the metric without recurring to the embedding?

Thank you in advance.

-artur palha