Curvature of a vector-bundle connection

Let $E\to M$ be a smooth vector bundle with connection $\mathrm{\nabla }$. For smooth vector fields $X,Y$ and a smooth section $s$ of $E$, the curvature of $\mathrm{\nabla }$ is

$R^{\mathrm{\nabla }}(X,Y)s:={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}s-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}s-{\mathrm{\nabla }}_{[X,Y]}s\mathrm{.}$

This uses the same sign convention as the curvature of an affine connection. The next proposition proves that this operator is tensorial and hence is an $\mathrm{End}(E)$-valued two-form. At this definition stage no metric, torsion, or bundle trivialization is assumed.

If $M$ is empty, $E$ has rank zero, or $M$ has dimension zero, the displayed operator is the unique zero curvature operator. For a rank-one bundle and for a manifold with boundary the same local formula applies without modification.