Covariant Derivative (Linear Connection)

Let $E \to M$ be a vector bundle. A covariant derivative on $E$ is a $\mathbb{K}$ -linear map

\nabla : C^\infty(E) \to C^\infty(T^*M \otimes E)

such that, for all $f \in C^\infty(M)$ and all $u \in C^\infty(E)$ , we have

\nabla(fu) = df \otimes u + f \nabla u

where $C^\infty(E)$ denotes the space of smooth sections of $E$ over $M$ .

Remind that

\text{Hom}(T^*M, E) \simeq C^\infty(T^*M \otimes E)

Therefore, $\nabla u$ has a more traditional view

\begin{aligned} &\nabla u : \text{Vect}(M) \to C^\infty(E) \\ &X \mapsto \nabla_X u \end{aligned}