Covariant derivative of a section

Let $E\to M$ be a vector bundle with a connection $\nabla$ . For a vector field $X$ on $M$ and a section $s\in\Gamma(E)$ , the connection produces another section $\nabla_X s$ .

Definition. The covariant derivative of $s$ along $X$ is the section

\nabla_X s \in \Gamma(E)

defined by the connection’s bilinear map $\nabla:\mathfrak X(M)\times\Gamma(E)\to\Gamma(E)$ . It is characterized by:

$C^\infty(M)$ -linearity in $X$ : $\nabla_{fX}s=f\,\nabla_X s$ ,
$\mathbb R$ -linearity in $s$ , and
the Leibniz rule $\nabla_X(fs)=X(f)s+f\nabla_Xs$ in the section slot.

Equivalent characterizations

Equivalently, viewing $\nabla s$ as a $T^*M\otimes E$ -valued object, one has $(\nabla_X s)(p)=(\nabla s)(p)(X_p)$ by contraction.

Examples

Trivial bundle: ordinary directional derivative. For $E=M\times\mathbb R^r$ with the trivial connection, $\nabla_X s$ is just the usual derivative of the vector-valued function $s$ in the direction $X$ .
Tangent bundle: covariant derivative of vector fields. For a connection on $TM$ , $\nabla_X Y$ is the covariant derivative of the vector field $Y$ along $X$ , recovering the classical Christoffel-symbol formula in coordinates.
Line bundle with connection 1-form. In a local trivialization of a complex line bundle with connection form $A$ , if $s=f\,e$ for a local frame $e$ , then $\nabla_X s = (Xf + A(X)\,f)\,e$ .