Recently within machine learning, there are a lot of works on non-convex optimization and natural gradients methods etc which are based on differential geometry, it gives rise to increased need to learn differential geometry in machine learning community.
After searching a bit, I found a possible routine might be:
John Lee's book series in the order of
Introduction to Topological Manifolds
-> Introduction to Smooth Manifolds
-> Riemannian Manifolds: An Introduction to Curvature
Is it a "reasonable" path to enable one to be able to get ideas and possibly to apply new results from differential geometry community(e.g. understand new papers/presentations etc. ), and if we follow it, is it still necessary to read do Carmo's two more texts that are mentioned a lot by many friends from math department. For practical reason, since machine learning research itself already takes majority amount of time, so that there is a preference not to read another books when it is not so necessary.