# 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 [itex](M,g)[itex] be a Riemannian manifold (or pseudo-Riemannian manifold) then an affine connection [itex]\nabla[itex] is Levi-Civita connection if it satisfy the following conditions

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

## Derivative along curve

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

Given a smooth curve [itex]\gamma[itex] on [itex](M,g)[itex] and a vector field [itex]V[itex] on [itex]\gamma[itex] its derivative is defined by

[itex]\frac{D}{dt}V=\nabla_{\dot\gamma(t)}V.[itex]

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