Dimension of the metric of a projection of a sphere

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Let ##(x_1,x_2,x_3)=\vec{r}(\theta,\phi)## the parametrization of a usual sphere.

If we consider a projection in two dimension ##(a,b)=\vec{f}(x_1,x_2,x_3)##

Then I don't understand how to use the metric, since it is ##g_{ij}=\langle \frac{\partial\vec{f}}{\partial x_i}|\frac{\partial\vec{f}}{\partial x_j}\rangle## which is a 3x3 matrix but we have only two coordinates ##a,b## in the projection.
 

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  • #2
WWGD
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AFAIK, you use the pullback metric from the polar coordinates. This is the way I have seen it done.
 

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