Among functions between manifolds those which transform probability distributions with densities to probability distributions with densities are characterised as almost submersions. As an application, conditions are given on sample size for sample means of distributions with density on Stiefel manifolds and Grassmannians to have densities.
"Density-Preserving Statistics and Densities for Sample Means." Ann. Probab. 6 (4) 688 - 694, August, 1978. https://doi.org/10.1214/aop/1176995490