Interesting theorem in differential geometry

Question

I was reading a while back in differential geometry, and noticed a theorem which states something along the lines of that for any given manifold there always exists a point where the metric tensor $g_{ij}$ is equal to $g^{i}_{j}$ and so the derivative with respect to any coordinate is just 0. There was a transformation involved, I believe it is called the exponential map, may someone please explain why that map was used?