In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). The fundamental theorem of Riemannian geometry states that there is unique connection which satisfies these properties.

In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection. The coordinate-space expression of the connection are called Christoffel symbols.

Formal definition

Let (M,g) be a Riemannian manifold (or pseudo-Riemannian manifold) then an affine connection is Levi-Civita connection if it satisfy the following conditions

1. Preserves metric, i.e., for any vector fields X, Y, Z we have , where Xg(Y,Z) denotes the derivative of function g(Y,Z) along vector field X.
2. Torsion-free, i.e., for any vector fields X and Y we have , where [X,Y] are the Lie brackets for vector fields X and Y.

Derivative along curve

Levi-Civita connection defines also a derivative along curves, usually denoted by D.

Given a smooth curve γ on (M,g) and a vector field V on γ its derivative is defined by

External link

• MathWorld: Levi-Civita Connection
• PlanetMath: Levi-Civita Connection