I agree that he may have gleaned something (possibly even a substantial something) from it. As I pointed out, it's easy to learn the applied parts of different

I agree that he may have gleaned something (possibly even a substantial something) from it. As I pointed out, it's easy to learn the applied parts of differential geometry in a day or two: curvature, fundamental forms, etc. are fairly simple, fairly intuitive ideas for someone with a good grasp of multivariate calculus and differential equations, particularly when you're restricting yourself to three dimensional manifolds. I'd note that that's different from having a rigorous understanding of arbitrary dimension differential geometry, and being able to rigorously show new (if simple/uninteresting) results. I wasn't trying to imply that you can't read a textbook (or any math text, for that matter) and not learn the material at as deep a level as anyone who's main reference is that text. It's a bit like reading someone's code after very heavy optimization, though: it's easy to miss little parts of how or why the algorithm works, and if you go over it once, without trying possible inputs/etc., then you're likely to miss something. Instruction and interpretation are like comments in code when you're dealing with specific proofs (they make it easier, but it isn't impossible without them). The thing that is harder (but not impossible) to get without some sort of feedback is a deep understanding of when a proof is rigorous and mathematical aesthetics.