 Affine connection

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.

An **affine connection** is a mathematical concept used primarily in differential geometry and the theory of manifolds. It provides a way to define a notion of parallel transport, which allows one to compare vectors at different points on a manifold. The affine connection also enables the definition of derivatives of vector fields along curves in a manifold.