In most physics situations one gets the metric as a positive definite symmetric matrix in some chosen local coordinate system.
Now if on the same space one has two such metrics given as matrices then how does one check whether they are genuinely different metrics or just Riemannian Isometries of each other (and hence some coordinate change can take one to the other).
If in the same coordinate system the two matrices are different then is it proof enough that they are not isometries of each other? (doing this test over say a set of local coordinate patches which cover the manifold)
Asked otherwise, given two “different" Riemannian Manifolds how does one prove the non-existence of a Riemannian isometry between them?
There have been two similar discussions on mathoverflow at this and this one.
And this article was linked from the later.
In one of the above discussions Kuperberg had alluded to a test for local isometry by checking if the Riemann-Christoffel curvatures are the same locally. If the base manifold of the two riemannian manifolds is the same then one can choose a common coordinate system in which to express both the given metrics and then given softwares like mathtensor by Mathew Headrick this is probably not a very hard test to do.
So can one simply patch up such a test through out a manifold to check if two given riemannian manifolds are globally isometric?
How does this compare to checking if the metrics are the same or not in a set of common coordinate patches covering the manifold?
I somehow couldn't figure out whether my query above is already getting answered by the above two discussions.
Best Answer
This goes by the name of the "equivalence problem" in riemannian geometry and it is an important problem in the classification effort of solutions to gravity field equations. Malcolm MacCallum has many results in this area. For example this paper from 1985 might be a place to start reading about it.