For the task of comparing distributions with statistical inference, nonparametric tests for common means and common covariances, e.g. principal component analysis (PCA), are popular.
Extending such concepts to non-Euclidean spaces we face four challenges
a) suitably defining PCA,
b) bringing PCs into a nested structure,
c) derive their asymptotics,
d) define (bootstrap) tests.
Among others, in the context of a) and b) we will see that spheres are statistically even more benign than Euclidean spaces.
And in the context of c) and d) we will encounter non-Euclidean limiting behaviors, so-called smeary and sticky limit theorems.