A connection on a vector bundle π:E->M is a way to "differentiate" bundle sections, in a way that is analogous to the exterior derivative d f of a function f. In particular, a connection del is a function from smooth sections Γ(M, E) to smooth sections of E with one-forms Γ(M, E⊗T^* M) that satisfies the following conditions. 1. del f s = s⊗d f + f del s (Leibniz rule), and 2. del s_1 + s_2 = del s_1 + del s_2.

bundle curvature | bundle section | curvature | Hermitian metric | Levi-Civita connection | parallel transport | principal bundle | second fundamental form