The Christoffel symbols are local coefficients of the Levi-Civita connection of a Riemannian metric geometry
and appear, for example, in the geodesic equation .
There are two closely related kinds, the first
kind second kind connection
in a chosen coordinate system ; they are not
tensor components, because their transformation law contains
second derivatives of the change of coordinates
(Meinrenken 2002, pp. 38-41).
It is always possible to pick a coordinate system on a Riemannian manifold such
that the Christoffel symbol vanishes at a chosen point .
In general relativity, Christoffel symbols are "gravitational forces,"
and the preferred coordinate system referred
to above would be one attached to a body in free fall.
See also Christoffel Symbol of the First Kind ,
Christoffel
Symbol of the Second Kind ,
Geodesic ,
Levi-Civita
Connection ,
Riemannian Geometry
Portions of this entry contributed by Todd
Rowland
Portions of this entry contributed by Eugene
Salamin
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References Meinrenken, E. Riemannian Geometry. Lecture notes, Spring 2002. https://www.math.toronto.edu/mein/teaching/LectureNotes/rieall.pdf . Misner,
C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation. Weinberg, S. Gravitation
and Cosmology: Principles and Applications of the General Theory of Relativity. Referenced on Wolfram|Alpha Christoffel Symbol
Cite this as:
Weisstein, Eric W. , with contributions by Todd Rowland and Eugene Salamin . "Christoffel
Symbol." From MathWorld https://mathworld.wolfram.com/ChristoffelSymbol.html
Subject classifications
. They are used to study its geometry
and appear, for example, in the geodesic equation.
There are two closely related kinds, the first
kind 
and the second kind 
. The second-kind coefficients specify the connection
in a chosen coordinate system; they are not
tensor components, because their transformation law contains
second derivatives of the change of coordinates
(Meinrenken 2002, pp. 38-41).
