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
- 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.
- Torsion-free, i.e., for any vector fields X and Y we have
![\nabla_XY-\nabla_YX=[X,Y]](a/../upload/math/a46e8900b1310f87838ba8dc11f1fa5f.png)
, 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
