Polar Coordinate Systems on Manifolds Is there agreement on how to interpret $r$ and $\varphi$ on a manifold if a reference point and a reference direction are given, or, put differently, does the definition of a reference point and, of a reference direction suffice to make polar functions on the manifold well defined?
While I think that the definition of $r$ as the geodesic distance from the reference point is commonly agreed, I'm having an uneasy feeling with t he definition of $\varphi$: is it measured as the fraction a of circle that is geodesically centered at the reference point  or, is the geodesic emanating from the reference point in the direction defined by $\varphi$?  
Maybe even the reference direction is agreed upon as being in a direction of principal curvature if it is uniquely defined.   
Any pointers to examples of polar coordinates on manifolds would be welcome. 
 A: Typically such a chart is only used or only useful when there is an SO(2)$\cong$U(1) isometry of the manifold, or more generally an SO($n$) isometry for a higher dimensional manifold. If so, then it's conventional to choose the angular coordinates in the standard way on the $(n-1)$-spheres which are the orbits under the group action. For example we would have $\varphi\in[0,2\pi)$ with identification $0\sim2\pi$ for $n=2$, rather than based on geodesic distance.
For the radial coordinate there are various useful choices. A very common choice is the "areal coordinate" $r_{\rm areal}$ or "circumferential coordinate" choice, which would put the (pseudo-)Riemannian metric on this sector of the geometry into the convenient form
$$
  ds^2 = r_{\rm areal}^2 d\Omega_{n-1} + \ldots
$$
where $d\Omega_{n-1}$ is the standard line element on the unit $(n-1)$-sphere (for example, $d\Omega_2=d\theta^2+\sin^2\theta d\phi^2$). This choice is to label the orbits (the spheres) by the radius of a Euclidean sphere with the same area (or circumference).
There may be other useful conventions, e.g. as you mentioned labeling spheres by their radial geodesic distance from the center. On a case-by-case basis, some of these choices are discussed in the general relativity literature (see e.g. Carroll, or MTW, or Hartle, or Schutz, or Wald, etc.).
A: It seems that the correct analog of polar coordinates is to be defined on a Riemannian manifold. So if $M$ is a manifold endowed with a Riemannian metric, and $x$ is a point in $M$, the exponential map $\exp_x : T_xM \to M$ is well-defined on a neighborhood of $0 \in T_xM$, and can be used a coordinate chart around $x$.
Since $T_xM$ is naturally an inner product space, we can speak about spheres in it. A sphere of radius $r$ in $T_xM$ will be mapped to a sphere of geodesic radius $r$, consisting of points at geodesic distance $r$ from $x$. That's the radial coordinate, and again it is well-defined for $r$ small enough.
If a direction $u \in T_xM$ is chosen, one can define the angle between a given point on a geodesic sphere of radius $r$ around $x$ and the given direction, again, using the exponential map: the point in question will be the image of a point $v$ lying on a sphere of radius $r$ in $T_xM$, and the angle between the point on the manifold and the chosen direction can just be measured as the angle in $T_xM$ between $u$ and $v$.
Proceeding in a similar manner, one can choose a collection of pairwise orthogonal vectors $u_1,\dots,u_{n-1}$ where $n = \dim M$, and measure angles in $T_xM$ between these directions and a given point, much like we do in Euclidean space. This gives meaning to polar coordinates around a point on a Riemannian manifold.
