I was having a discussion with my colleague, and he posited that:
"If one only has access to the metric tensor, is this enough information to totally determine (or recover) the original (Riemann) manifold shape?"
I want to say no, but I can't think of any counter examples. From what the maths (as I understand it) says:
(i) The metric tensor helps to define local distances and angles as one moves around the manifold, and
(ii) The Riemann curvature tensor can be expressed in terms of the metric tensor
And so from this information, is it not sufficient to only possess the metric tensor in order to fully determine manifold global shape? Or am I missing something key here?