Curvature of a vector bundle connection

Let $E\to M$ be a vector bundle with a connection on a vector bundle $\nabla$ . Let $X,Y$ be smooth vector fields on $M$ , and let $s$ be a smooth section of $E$ .

Definition. The curvature of $\nabla$ is the map $R^\nabla$ defined by

R^\nabla(X,Y)s \;:=\; \nabla_X\nabla_Y s \;-\; \nabla_Y\nabla_X s \;-\; \nabla_{[X,Y]} s,

where $[X,Y]$ is the Lie bracket of vector fields.

For each $X,Y$ , the operator $R^\nabla(X,Y):\Gamma(E)\to\Gamma(E)$ is $C^\infty(M)$ -linear in $s$ , hence defines a bundle endomorphism of $E$ . Moreover, $R^\nabla$ is $C^\infty(M)$ -linear in $X$ and $Y$ and skew-symmetric, so it can be viewed as an $\mathrm{End}(E)$ -valued bundle-valued differential 2-form on $M$ . In a local frame, it is represented by the curvature 2-form matrix.

Examples

Trivial connection is flat. For the trivial connection on $M\times\mathbb R^r$ , mixed derivatives commute and $R^\nabla=0$ .
Levi-Civita curvature. For the Levi-Civita connection on $TM$ , $R^\nabla$ is the Riemann curvature tensor (in $(1,3)$ form), encoding sectional curvature and holonomy phenomena.
Constant matrix connection on a trivial bundle. On $U\subset\mathbb R^n$ , take $\nabla = d + \sum_k A_k\,dx^k$ with constant matrices $A_k$ . Then $R^\nabla$ corresponds to the commutators $[A_k,A_\ell]$ ; it vanishes exactly when the matrices $A_k$ pairwise commute.