I'll be doing an independent study with one of my profs in differential geometry next semester (my university did not happen to offer an intro diff. geometry course next semester like it usually does). I'll be mainly working out of Barrett O'Neill's book but will also be checking out different perspectives by looking at do Carmo's book (and maybe Spivak's?) I've been planning out the rest of my semesters and even if I end up taking courses in a wide range of branches in mathematics, I'll still have quite a bit of free credits to delve more deeply into one subject. If I do choose to go further into differential geometry, what are some important classes to take/books to read? Books I've looked into so far are Do Carmo's Riemannian Geometry, Barrett O'neill's Semi-Riemannian Geometry, as well as differential topology books like Milnor's topology from a differentiable viewpoint or Lee's introduction to smooth manifolds (I understand these are important for more advanced work in differential geometry?) What is the recommended order I should learn these subjects in? Any other suggestions/recommendations?