Affine connection on a smooth manifold

An affine connection on a smooth manifold $M$ is a connection on the tangent bundle $TM$ in the sense of Connection on a smooth vector bundle. Thus it assigns a vector field ${\mathrm{\nabla }}_{X}Y$ to two vector fields and is function-linear in the differentiating direction $X$, real-linear in $Y$, and obeys ${\mathrm{\nabla }}_X(fY)=X(f)Y+f{\mathrm{\nabla }}_{X}Y$.

The word affine imposes neither a metric nor torsion freeness. The tangent-bundle specialization allows the Lie bracket of $X,Y$ to be compared with their covariant derivatives; there is no corresponding bracket on sections of a general vector bundle. Manifolds with boundary are allowed, and tangent vectors at their boundary need not be tangent to the boundary. The empty and zero-dimensional manifolds have the unique tangent-bundle connection.