Levi-Civita connection

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 <math>(M,g)</math> be a Riemannian manifold (or pseudo-Riemannian manifold) then an affine connection <math>\nabla</math> is Levi-Civita connection if it satisfy the following conditions

Preserves metric, i.e., for any vector fields <math>X</math>, <math>Y</math>, <math>Z</math> we have <math>Xg(Y,Z)=g(\nabla_X Y,Z)+g(Y,\nabla_X Z)</math>, where <math>Xg(Y,Z)</math> denotes the derivative of function <math>g(Y,Z)</math> along vector field <math>X</math>.
Torsion-free, i.e., for any vector fields <math>X</math> and <math>Y</math> we have <math>\nabla_XY-\nabla_YX=[X,Y]</math>, where <math>[X,Y]</math> are the Lie brackets for vector fields <math>X</math> and <math>Y</math>.

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

Given a smooth curve <math>\gamma</math> on <math>(M,g)</math> and a vector field <math>V</math> on <math>\gamma</math> its derivative is defined by

<math>\frac{D}{dt}V=\nabla_{\dot\gamma(t)}V.</math>