I've been reading about probability on manifolds. What bothers me is that there's no clear way to generalize the Gaussian distribution to manifolds. If we instead assign a probability distribution to each tangent space, and make sure these probability distributions vary smoothly, then this gives us a very clear way to generalize probability distributions to manifolds. Does such a concept exist in literature?

I want to emphasize I'm being very loose with my definitions. What does it mean to assign a probability distribution to a manifold? Well, I suppose for each tangent space $T_xM$, we assign to it a probability distribution $p_x :T_xM \to [0,\infty)$ and $p_x$ "varies smoothly" for $x \in M$. Please understand, this is a soft question and I am being loose with my definitions since I am trying to get a better idea of what's going on.

Any suggestions of papers or textbooks, I would greatly appreciate.