What do I need to learn to start on the rigorous study of differential geometry? I'm about to start my 3rd undergrad year at school, and have taken Cal 1-3, Linear Algebra, Elementary Number Theory, and ODEs.
If I want to take the most direct path to studying differential geometry, what should I start learning now? Should I start on Real Analysis or Topology? Should I just pick up Spivak's Intro to Differential Geometry? My multivariable calculus class did not cover differential forms, so should I read his Calculus on Manifolds or Munkres' Analysis on Manifolds first?
I'm pretty sure that I want to study differential geometry in grad school and I'd like to start on it before then (ideally, I'd like to be able to publish something -- not necessarily extremely deep -- in diff geo before applying to grad schools).
So what path should I take to self-study it?
NOTE: My school has 1 undergrad course in Differential Geometry available in the Spring, so I'll probably do that next year, but I'd rather start now and go well beyond whatever I'm going to learn in that class.