Affine connection ($\nabla$)

An affine connection on a smooth manifold is a connection on a vector bundle for its tangent bundle. It satisfies ${\nabla }_{fX}Y=f{\nabla }_{X}Y$ and ${\nabla }_X(fY)=X(f)Y+f{\nabla }_{X}Y$. Its curvature isThe 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.

Connections ${\nabla }^1,{\nabla }^2$ on $T{M}_1,T{M}_2$ give a connection onby taking the direct sum of their pullback connections. In product coordinates its Christoffel symbols have the two factor blocks and zero mixed blocks. On arbitrary fields, the ordinary derivatives of their coefficients are still taken in both factors. On fields lifted separately from the factors it satisfies ${\nabla }_{Y_1+Y_2}(X_1+X_2)={\nabla }_{Y_1}^1{X}_1+{\nabla }_{Y_2}^2{X}_2$.

For a product affine connection, its curvature satisfiesExpand the defining derivative commutator on lifted vector fields. Opposite-factor fields commute and have zero mixed covariant derivatives, leaving exactly the two factor curvatures. Tensoriality gives the formula for arbitrary tangent vectors. Thus the product connection is flat if the two factors are flat; for a product Riemannian metric this applies to its Levi-Civita connection.