Meaningfulness of metric and covariant derivative induced by spherical coordinations

Question

On $S^2$ have the spherical parametrization $f:(\theta,\phi)\rightarrow (\sin(\theta) \cos(\phi), \sin(\theta) \sin(\phi), \cos(\theta))$. Is it meaningful to talk about the Riemannian metric induced by this only parameterisation? As far as I know we can define a Riemannian metric on any manifold induced by all parametrizations with partition of unity, not only with one parametrization.

Answer 1

This $f$ is an embedding of $(0,\pi) \times (0,2\pi)$ into $S^2$. You can use this to transport the Euclidean metric to the image of $f$ in $S^2$, which is the sphere minus a meridian. But this does not give you a metric on all of $S^2$.