Christoffel symbols

A vector field in n can be seen as a differentiableMathworldPlanetmathPlanetmath (C) map V:nn.

Or as a sectionMathworldPlanetmath nVT(n) where Tnn×n is the n’s trivial tangent bundle obeying p(p,V(p)Tp(n)) with Tp(n)n being the tangent spacePlanetmathPlanetmath at p.

Another viewpoint about tangent vectors is that they are also linear operators called derivationsPlanetmathPlanetmath and they act over scalars f:n via pVf|p=V(p)f|p.

Let X be one of them and dX|p its Jacobian matrix evaluated at the point pn. Then, for any other vector field Y:nn,


measures how X varies in the direction Y at p.

We have dX|p(Y(p))=(Y(p)X1|p,,Y(p)Xn|p), where X=sXses in componentsPlanetmathPlanetmath. Also, it is obvious that pdX|p(Y(p)) defines a new vector field in n which is symbolized as


We can be consider it as a bilinear map


Further, it is easy to see that for any scalar f:n

  1. 1.


  2. 2.


  3. 3.


  4. 4.


Here we have abbreviated (as usual) Yf=YF and the operationMathworldPlanetmath [X,Y] is the Lie bracketMathworldPlanetmath.

This D is called the standard connectionMathworldPlanetmath of n.

Now, let M be a n-dimensional differentiable manifold and let TM be its tangent bundle. The set of differentiable sections Γ(M)={X:MTM} is a differentiable Lie algebra which is endowed with a differentiable inner product g:Γ(M)×Γ(M) via


in each Tp(M)n.

It is possible construct a bilinear operator


compatibleMathworldPlanetmath with g and which satisfies the following properties

  1. 1.


  2. 2.


  3. 3.


  4. 4.


The Fundamental Theorem of Riemannian Geometry establishes that this exists and it is unique, and it is called the Levi-Civita connectionMathworldPlanetmath for the metric g on M.

Now, if one uses a coordinated patch in M one has a set of n-coordinated vector fields 1,..,n meaning i=ui being ui the coordinate functions. These are also dubbed holonomic derivations.

So it makes sense to speak about the derivativesPlanetmathPlanetmath ij and since the i are tangentPlanetmathPlanetmathPlanetmath which generate at a point Tp(M), then ij is also tangent, so there are n×n numbers (functions if one varies position) Γijs which enters in the relationMathworldPlanetmath


These coefficientsMathworldPlanetmath Γijs are called Christoffel symbolsMathworldPlanetmath and an easy calculation shows that


where gij=g(i,j), gij are the entries of the matrix [gij]-1 and gij,k=k(gij).

Routinely one can check that under a change of coordinates uiwj these functions transform as


here we have used Einstein’s sum convention (m,n,p-sums) and the term


shows that the Γkli are not tensors.

For a proof please see the last part in:

Connection with base vectors.

Let us assume that coordinates ui are referred to a right-handed orthogonalMathworldPlanetmathPlanetmathPlanetmath Cartesian system with attached constant base vectors 𝐞i𝐞i and coordinates wj referred to a general curvilinear system attached to a local covariant base vectors 𝐠j and local contravariant base vectors 𝐠k, both systems embedded in the Euclidean space n. We shall also suppose diffeomorphic the transfomation uiwj. Then, by definition

𝐠j:=uiwj𝐞i,𝐠j:=wjui𝐞i, (1)

and its inversesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath

𝐞i=𝐞i=uiwj𝐠j=wjui𝐠j. (2)

Let us consider differentiationMathworldPlanetmath of base vectors 𝐠j, which may be written from (1),(2)


and using the Christoffel symbols this becomes

𝐠jwk=Γjks𝐠s=Γjkr𝐠r, (3)


Γjks=2uiwjwkuiws,Γjkr=grsΓjks. (4)

Since the transformationMathworldPlanetmath of covariant and contravariant metric tensorsMathworldPlanetmath are given by


is easy to see from here that Christoffel symbol Γjks enjoy the property

Γjks=12(gjswk+gkswj-gjkws) (5)

In a similarMathworldPlanetmathPlanetmathPlanetmath way we find for the derivative of the contravariant base vectors

𝐠jwk=-Γksj𝐠s. (6)

Is easy to show the following results:


comma denoting differentiation with respect to the curvilinear coordinates wj and g=|gjk|. When the coordinate curves are orthogonal we have the following formulae for the Christoffel symbols: (repeated indices are not to be summed)

Title Christoffel symbols
Canonical name ChristoffelSymbols
Date of creation 2013-03-22 15:43:52
Last modified on 2013-03-22 15:43:52
Owner juanman (12619)
Last modified by juanman (12619)
Numerical id 24
Author juanman (12619)
Entry type Definition
Classification msc 53B20
Classification msc 53-01
Synonym connection coefficients
Related topic Connection