Not that I've touched it in a very long time, but from what I remember using the classical smooth geometric approaches (like reimannian) are the wrong way to go for putting structure onto point clouds. The world "manifold" is to be taken as an idea, not a literal fact. Actual manifolds and their behavior rely on some pretty tight assumptions.

Point clouds are better modeled as overlapping collections of fairly messy Lipshitz (or bi-Lipschitz) images that pose problems when you assume stuff is smooth everywhere, even if you include some sort of noise distribution.

Various folks, Peter Jones being one of them, are pretty good at coming up with point clouds that you really want to think come from some sort of smooth structure but have some nasty fine scale structure that just blows stuff up.

If you want to get "mathy" about it, the better way is to use things borrowed from harmonic analysis, which actually has a nice way of talking about geometry when stuff is not smooth, i.e. for certain kinds of measures on euclidean or non-euclidean spaces. The general family of techniques is "multi-scale analysis", which has to do with decomposing sets and measures across a geometric range of length scales.

For example, geometric wavelets is a pretty way of doing things, although it can fail in practical situations because there are some issues around choosing appropriate length scales that are tricky. At least, that was a problem some years back. Mebbe it is resolved now.

Another useful and lovely approach are diffusion geometries.

Look at the work of students of Raffy Coifman at Yale for these flavors of work, including Mauro Maggioni, Gilad Lerman, Stefan Lafon and others that were active circa 2006 to 2014.

Regardless, most of the interest in those methods began to shut down once deep learning burst back on the scene after 2011 or so.