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

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 in the section slot (see Leibniz rule for a connection ).

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

1. 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 $X$.
2. 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.
3. 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 $\nabla_X s = (Xf + A(X)\,f)\,e$.