I am having a confusion regarding the type of geometry of a coordinate system, whether it is Euclidean or not. Intuitively if we consider the spherical polar coordinates, it is a Euclidean geometry since the sphere with unspecified radius is embedded in $\mathbb{R}^3$, but at the same time if we take the $S^2$ then it is locally a two dim space $\theta,\phi$ are specified. This is a non-Euclidean geometry globally, but locally Euclidean. If we see the form of the metric, for spherical polar coordinate, $$ ds^2=dr^2+r^2d\theta^2+r^2sin^2 \theta d\phi^2$$ which is a Euclidean metric. Now for the surface if $S^2$, the metric reads $$ ds^2=d\theta^2+sin^2\theta d\phi^2$$ How to see this metric corresponds to Non-Euclidean geometry?
One way to see (not incorporating the curvature tensor) is we would be able to find a linear transformation such a way the metric in Euclidean geometry may be reduced to the standard $\delta_{ij}$. This is not possible with non Eulcidean geometry. But how to find the transformation for the metric in spherical polar coordinates?
and, if we find out the diffrence between two points infinitesimally close, or globally separated, the later is non Euclidean geometry. But the metric is same for both the cases I assume. Does that mean from the metric we can not find out the underlying geometry?