Dimension of the metric of a projection of a sphere

[jk22]

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.

 

[WWGD]

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