So as a quick background, I’m an applied mathematics PhD student whose work is in model reduction, I’ve only a sparse background in geometry and topology, but the problem I’m working is formulated as a system of PDEs, and in projecting it to some reduced model, it can be formulated as an ODE system on a non-Euclidean space determined by some data analysis. I’d like to assume this space has some degree of smoothness, but not necessarily infinitely smooth.

To understand the notion of distance and inner product as I project onto this “data-driven” manifold, I’m wondering how to geometrically interpret distance and paths on manifolds with a finite degree of smoothness. Is there a notion of precise distance/path-length for curves on manifolds with a finite amount of smoothness?

Again, I realize that I may be asking a question that makes little to no sense as I’m ignorant of a lot of differential geometry; I picked up O’Neill and Lee’s Smooth Manifolds from the library to familiarize myself, but I’m wondering if I can get a little generality here; using empirical data to glean an underlying manifold doesn’t guarantee infinite smoothness.