= Affine connection
{title2=$\nabla$}
{wiki}
= Linear connection on a manifold
{synonym}
An affine connection on a <smooth manifold> is a <connection on a vector bundle> for its <tangent bundle>. It satisfies $\nabla_{fX}Y=f\nabla_XY$ and $\nabla_X(fY)=X(f)Y+f\nabla_XY$. Its curvature is
$$
R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z.
$$
The <Leibniz rule> makes curvature tensorial in all three slots. An affine connection is also called a linear connection on the manifold; a <Levi-Civita connection> is the distinguished torsion-free metric-compatible example.
Back to article page
