Curvature of an affine connection

Let $M$ be a smooth manifold, let $\mathrm{\nabla }$ be an affine connection on $M$ in the sense of Affine connection on a smooth manifold, and let $X,Y,Z$ be smooth vector fields. With the sign convention used throughout this page, the curvature of $\mathrm{\nabla }$ is

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

where $[X,Y]$ is the bracket of The Lie bracket of smooth vector fields. We usually write $R$ when the connection is understood. The connection is flat or curvature-free when $R(X,Y)Z=0$ for every triple of smooth vector fields.

The bracket correction is part of the definition. The raw commutator of two covariant derivatives is not function-linear in its differentiating fields. No metric or torsion hypothesis is imposed here. On an empty or zero-dimensional manifold the condition is vacuous; on manifolds with boundary the same formula uses the ambient tangent bundle supplied by the definition of an affine connection.