Classes of coordinate-invariant omnibus goodness-of-fit tests on compact Riemannian manifolds are proposed. The tests are based on Giné’s Sobolev tests of uniformity. A condition for consistency is given. The tests are illustrated by an example on the rotation group SO(3).
"Sobolev tests of goodness of fit of distributions on compact Riemannian manifolds." Ann. Statist. 33 (6) 2957 - 2966, December 2005. https://doi.org/10.1214/009053605000000697