Levi-Civita connection

From Academic Kids

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

  1. 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>.
  2. 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>.

Derivative along curve

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>

External link

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