Pretty much what the question says. Right. So, to remind everyone, spherical coordinates are:
\begin{align} x & = \rho \sin(\phi)\cos(\theta) \\ y & = \rho \sin(\phi)\sin(\theta) \\ z & = \rho \cos(\phi) \\ \rho^2 & = x^2 +y^2 +z^2 \end{align}
After computing the pullback, I get the result $d\rho^2 + \rho^2 \sin^2(\phi) d \phi ^2 + \rho^2 d\theta^2 $
(Question 1: Is this computation correct? Is there an easier way to calculate this?)
What is the geometric intuition here? If I had a tangent vector $v\in T_pM$ does moving by $\rho$ tell me that I have to adjust my $\phi $ and $\theta$ in my tangent vector? My own intuition tells me that this change of coordinates allows for a sort of different expression of "straightness". Rotating by $\theta$ is very natural in these coordinates but to express a nonradial line in spherical coordinates sounds tricky. So that is how I am sort of justifying this to myself. What are others' intuition on this pullback metric business?