I'm finishing my PhD in applied math (control theory but my interest is more general than just to applications in control) and I want to study Riemannian geometry, the catch is that all my "knowledge" of differential geometry is from my bachelor in physics.
I'm planning using Lee's Introduction to Smooth Manifolds. In the prerequisites it says
This subject draws on most of the topics that are covered in a typical undergraduate mathematics education. The appendices (which most readers should read, or at least skim, first) contain a cursory summary of prerequisite material on topology, linear algebra, calculus, and differential equations.
It is a good path? Should I add a supplementary differential geometry first?
John Lee's "Introduction to Smooth Manifolds" is the second entry in his trilogy of graduate/advanced-undergraduate level differential topology and geometry texts.
Since you almost certainly have sufficient experience with linear algebra, calculus, and differential equations, I highly recommend starting with the first text: "Introduction to Topological Manifolds". Even if you are already strong in topology, check out the table of contents and make sure you know everything in this text, as it will be assumed knowledge for "Smooth Manifolds".
Additionally, if your goal is become proficient in Riemannian geometry, the second text will not be sufficient, and you will want to follow-up with the third text: "Introduction to Riemannian Manifolds".
It's a bit of a time investment to read three texts, though. If you're impatient, you can try jumping into "Riemannian Manifolds" and see if your existing knowledge of DG can carry you. If you start hitting walls, though, you now know where to go to fill those gaps.