What is different from geodesic polar coordinates and other polar coordinates?
Geodesic polar coordinates has a form of $$ds^2=dr^2+f(r,\theta)^2\,\,d\theta^2$$
In $S^2$, $f(r,\theta)=\sin(r)$ which reduces to \begin{align} ds^2=dr^2+\sin(r)^2 d\theta^2 \end{align} Here $r$ and $\theta$ are angle coordinates,
In spherical coordinates we know, orthonormal frame \begin{align} ds^2=dr^2 +r^2 d\theta^2 +r^2\sin(\theta)^2 d\phi^2 \end{align} Here $r$ is radial and $\theta$, $\phi$ are polar, azimuthal angle. $\theta \in [0,\pi]$, $\pi \in [0,2\pi]$
We can simply set sphere to be constant radius $i.e$ $r=1$, so that this metric reduces to the form \begin{align} ds^2=d\theta^2+\sin(\theta)^2 d\phi^2 \end{align} Actually it is same with geodesic polar coordinates in $S^2$.
Here i am confused with terminology geodesic polar coordinates?
I want to know the exact definition of geodesic polar coordinates.
I know geodesic normal coordinate is defined by, for its coordinate, $p$, given metric's first order derivative is vanishes. $i.e$ $\partial g |_p=0$