Connection on a vector bundle

Remarks

Example: on the trivial bundle $E = M \times \R^r$, a section is a smooth map $s = (s^1, \dots, s^r)$, and $\nabla_X s = (X s^1, \dots, X s^r)$ is a connection; clause 3 is the product rule $X(f s^i) = (Xf) s^i + f X s^i$. Non-example: on the tangent bundle, the map $(X, Y) \mapsto [X, Y]$ satisfies clause 2 but fails clause 1, since $[fX, Y] = f[X, Y] - (Yf) X$. By clause 1, $(\nabla_X s)(p)$ depends only on $X_p$, so $\nabla$ can be seen as a map $\Gamma(E) \to \Gamma(T^*M \otimes E)$. In a local frame $e_1, \dots, e_r$ of $E$, $\nabla e_j = \sum_i \omega^i_j \otimes e_i$ for a matrix $(\omega^i_j)$ of $1$-forms, and locally $\nabla = d + \omega$. The difference of two connections is $C^\infty(M)$-linear in both arguments, so it is a section of $T^*M \otimes \mathrm{End}(E)$; thus connections form an affine space over this space of sections. Complex vector bundles are treated the same way with complex scalars.