# Meaningfulness of metric and covariant derivative induced by spherical coordinations

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 paramatrizations with partition of unity, not only with one parametrization.