Suppose I am working with a Riemannian manifold $(M,g)$, and I have a particular coordinate expression for the metric $g$.
What topological information can I infer about the manifold $M$?
For example ($S^3$ with Hopf coordinates), suppose I have coordinates $(\eta, \xi_1 , \xi_2 ) $ in which the metric takes the form:
$ds^2 = d \eta^2 + \sin^2(\eta) d \xi_1^2 + \cos^2(\eta) d\xi_2^2$
for $0 < \eta < \pi/2$, and $0 < \xi_1 , \xi_2 < 2 \pi$. If I didn't already know this was a metric for $S^3$, how could I work that out? How do I know this isn't a metric for another three manifold, say $S^2 \times S^1$? Are there topological invariants I can compute from the metric to distinguish between, say, these two possibilities?
EDIT: Changed the inequalities to be strict so that the metric in my example doesn't degenerate.
The expression you wrote does not define a Riemannian metric for coordinates on $S^3$, because when $\eta=0$ or $\pi/2$ the metric degenerates. It does define coordinates for a portion of $S^3$, namely the portion where the coordinates satisfy the strict inequalities $0 < \eta < \pi/2$, $0 < \xi_1, \xi_2 < 2 \pi$.
So perhaps you meant to use strict inequalities $0 < \eta < \pi/2$, $0 < \xi_1, \xi_2 < 2 \pi$? In which case I would say that the expression you wrote also defines coordinates for another 3-manifold, namely $$(0,\pi/2) \times (0,2\pi) \times (0,2\pi) $$