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.