Two families of invariant tests for independence of random variables on compact Riemannian manifolds are proposed and studied. The tests are based on Gine's Sobolev norms which are obtained by mapping the manifolds into Hilbert spaces. For general compact manifolds, randomization tests are suggested. For the bivariate circular case, distribution-free tests based on uniform scores are considered.
"Sobolev Tests for Independence of Directions." Ann. Statist. 13 (3) 1140 - 1155, September, 1985. https://doi.org/10.1214/aos/1176349661