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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.

Bernard
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    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 $S^2$, but this doesn't give you a metric on all of $S^2$. – Alex Provost Dec 31 '18 at 23:56
  • Thanks, that solved my problem –  Jan 01 '19 at 08:32
  • In that case I have made the comment into an answer, in order to remove it from the unanswered list. – Alex Provost Jan 01 '19 at 16:06

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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 $S^2$.

Alex Provost
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