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. 



Polar Coordinate Systems on Manifolds
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.).


