This is maybe late for your seminar, but a classical textbook about Riemannian geometry, including relations between curvature and metric tensor, with any signature, is
Riemannian geometry by L.P.Eisenhart
Another very interesting book for you could be
Spaces of constant curvature by J. A. Wolf
Indeed, it seems that you are searching for the Riemannian manifolds whose metric element can be written as sum and/or difference of squares of coordinate differentials. This implies that the curvature is constant and equal to 0. As shown in Wolfs'book, this can be locally realized by several manifolds with different "global geometry". As example in dimension 2, $ds^2=dx^2+dy^2$ can be realized on the (Euclidean) plane, on the cylinder, on the torus, on the Moebius strip and on the Klein bottle, while $ds^2=dx^2-dy^2$ on the (Minkowski) plane, on the torus and on the Klein bottle.