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

(M,g)

\nabla

is Levi-Civita connection if it satisfy the following conditions

Preserves metric, i.e., for any vector fields $X$ , $Y$ , $Z$ we have $Xg(Y,Z)=g(\nabla_X Y,Z)+g(Y,\nabla_X Z)$ , 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 $\nabla_XY-\nabla_YX= X,Y$ , where $X,Y$ are the Lie brackets for vector fields $X$ and $Y$ .

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

D

. Given a smooth curve

\gamma

(M,g)

V

\gamma

its derivative is defined by

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

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