Notes on Vector Bundles

−→ V, (v1, v2) −→ v1+v2, are smooth. Note that we can add v1, v2∈V only if they lie in the same fiber over M , i.e. π(v1)=π(v2) ⇐⇒ (v1, v2) ∈ V ×M V. The space V ×M V is a smooth submanifold of V ×V , as follows immediately from the Implicit Function Theorem or can be seen directly. The local triviality condition means that for every point m∈M there exist a neighborhood U of m in M and a diffeomorphism h : V |U ≡ π (U) −→ U×R, such that h takes every fiber of π to the corresponding fiber of the projection map π1 : U×R −→U , i.e. π1◦h=π on V |U so that the diagram