Levi-Civita connection

On any Riemannian manifoldMathworldPlanetmath M,g, there is a unique torsion-free affine connectionMathworldPlanetmath on the tangent bundle of M such that the covariant derivativeMathworldPlanetmath of the metric tensor g is zero, i.e. g is covariantly constant. This condition can be also be expressed in terms of the inner product operation ,:TM×TM induced by g as follows: For all vector fields X,Y,ZTM, one has




This connection is called the Levi-Civita connectionMathworldPlanetmath.

In local coordinates {x1,,xn}, the Christoffel symbolsMathworldPlanetmath (http://planetmath.org/Connection) Γjki are determined by

Title Levi-Civita connection
Canonical name LeviCivitaConnection
Date of creation 2013-03-22 13:59:25
Last modified on 2013-03-22 13:59:25
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 7
Author rspuzio (6075)
Entry type Definition
Classification msc 53B05