I have a hazy notion of some stuff in differential geometry and a better, but still not quite rigorous understanding of basics of differential topology.
I have decided to fix this lacuna once for all. Unfortunately I cannot attend a course right now. I must teach myself all the stuff by reading books.
Towards this purpose I want to know what are the most important basic theorems in differential geometry and differential topology. For a start, for differential topology, I think I must read Stokes' theorem and de Rham theorem with complete proofs.
Differential geometry is a bit more difficult. What is a connection? Which notion should I use? I want to know about parallel transport and holonomy. What are the most important and basic theorems here? Are there concise books which can teach me the stuff faster than the voluminous Spivak books?
Also finally I want to read into some algebraic geometry and Hodge/Kähler stuff.
Suggestions about important theorems and concepts to learn, and book references, will be most helpful.