Скачать или смотреть Introducing Vector Bundles

Vector bundles are one of the most important concepts of differential geometry, and of course I'm not doing them justice. I only introduce some of the more fundamental constructions.


The big theorem I mention about sections of vector bundles being projective modules, is called the (Serre-)Swan theorem.


Fun fact: the smooth functions R on a manifold also form a vector bundle L (the trivial line bundle), just like every commutative ring is also a projective module over itself. Since a k-vector space V is always isomorphic to the vector space Hom(k, V), we can represent the sections functor Γ: Vect(M) → R-Mod by the functor Mor(L, -). In other words: the sections of a vector bundle E are precisely the bundle morphisms L → E.