Vector Bundles and Locally Free Sheaves

Abstract

Let p: E → M be a vector bundle of rank r = dim F < ∞. The smooth sections of the bundle E over U ⊂ M form the vector space C ^∞(U, E); let us recall that the section s over U is the mapping s: U → M such that p o s = id _U . Obviously, the family U →C ^∞ (U, E), where U runs through open sets in the manifold M possesses the structure of a sheaf. If V ⊂ U, then there exists a mapping ρ ^ν _u : (U, E) →(V, E) of restriction of sections over U to sections over V. This sheaf is denoted by C ^∞_E and (U, E) by C ^∞_E (U) in order to be in agreement with notations used in the definition of sheaves.