On a Riemannian manifold M, there is a canonical connection called the Levi-Civita connection (pronounced lē-vē shi-vit-ə), sometimes also known as the Riemannian connection or covariant derivative. As a connection on the tangent bundle, it provides a well-defined method for differentiating vector fields, forms, or any other kind of tensor. The theorem asserting the existence of the Levi-Civita connection, which is the unique torsion-free connection del on the tangent bundle TM compatible with the metric, is called the fundamental theorem of Riemannian geometry.

Christoffel symbol | covariant derivative | curvature | fundamental theorem of Riemannian geometry | geodesic | principal bundle | Riemannian manifold | Riemannian metric