Low dimensional manifold in a high dimensional space and Geodesic distance

It is a common assumption that high-dimensional objects are lying in low-dimensional manifolds. And this constitutes a foundation for manifold learning or dimensional reduction techniques or (a way to beat the curse of dimensionality).

My question is that assuming this is valid, how one can utilize this assumption in doing something such as manifold learning?

I think the general goal is to find a nonlinear representation of this high-dimensional objective using a small degree of freedom.

However, we know neither the dimensionality of the tentative low-dimensional manifold nor the corresponding Geodesic distance. It seems to me that there is no effective way to achieve the goal of finding the nonlinear representation of the high-dimensional objective.

I would like to know if I am missing something or this manifold learning is just an engineering approach as an empirical method that works some cases but not all the time, without any theoretical understanding.

Any comments/suggestions/answers will be very appreciated. Thank you in advance.