PlanetMath: Christoffel symbols

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Christoffel symbols

(Definition)

A vector field in ${\mathbb{R}}^n$ can be seen as a differentiable ( $C^{\infty}$ ) map $V\colon{\Bbb{R}}^n\to {\mathbb{R}}^n$ .

Or as a section ${\mathbb{R}}^n\stackrel{V}\to T({\mathbb{R}}^n)$ where $T{\mathbb{R}}^n\equiv{\mathbb{R}}^n\times{\mathbb{R}}^n$ is the ${\mathbb{R}}^n$ 's trivial tangent bundle obeying $p\mapsto (p,V(p)\in T_p({\mathbb{R}}^n))$ with $T_p({\mathbb{R}}^n)\equiv{\mathbb{R}}^n$ being the tangent space at .

Another viewpoint about tangent vectors is that they are also linear operators called derivations and they act over scalars $f\colon {\mathbb{R}}^n\to{\mathbb{R}}$ via $p\mapsto Vf\vert _p=V(p)\cdot\nabla f\vert _p$ .

Let be one of them and $dX\vert _p$ its Jacobian matrix evaluated at the point $p\in{\Bbb{R}}^n$ . Then, for any other vector field $Y\colon{\mathbb{R}}^n\to{\mathbb{R}}^n$ ,

$\displaystyle dX\vert _p(Y(p))$

measures how

varies in the direction

We have $dX\vert _p(Y(p))=(Y(p)\cdot\nabla X^1\vert _p,...,Y(p)\cdot\nabla X^n\vert _p)$ , where $X=\sum_sX^se_s$ in components. Also, it is obvious that $p\mapsto dX\vert _p(Y(p))$ defines a new vector field in ${\mathbb{R}}^n$ which is symbolized as

$\displaystyle D_YX$

We can be consider it as a bilinear map

$\displaystyle D:T({\mathbb{R}}^n)\times T({\mathbb{R}}^n)\to T({\mathbb{R}}^n).$

$\displaystyle (X,Y)\mapsto D_XY$

Further, it is easy to see that for any scalar $f\colon {\mathbb{R}}^n\to{\mathbb{R}}$

$D_{fY}X=fD_YX$
$X(Y\cdot Z)=D_XY\cdot Z+X\cdot D_XZ$

Here we have abbreviated (as usual) $Yf=Y\cdot\nabla F$ and the operation

is the Lie bracket.

This is called the standard connection of ${\mathbb{R}}^n$ .

Now, let be a n-dimensional differentiable manifold and let be its tangent bundle. The set of differentiable sections $\Gamma(M)=\{X\colon M\to TM\}$ is a differentiable Lie algebra which is endowed with a differentiable inner product $g\colon\Gamma(M)\times\Gamma(M)\to{\mathbb{R}}$ via

$\displaystyle g(X,Y)\vert _p=X(p)\cdot Y(p)$

in each $T_p(M)\equiv{\mathbb{R}}^n$ .

It is possible construct a bilinear operator $\nabla$

$\displaystyle \nabla\colon \Gamma(M)\times\Gamma(M)\to\Gamma(M)$

compatible with

and which satisfies the following properties

$\nabla_{fY}X=f\nabla_YX$
$\nabla_Y(fX)=(Yf)X+ f\nabla_YX$
$\nabla_XY-\nabla_YX=[X,Y]$
$Xg(Y,Z)=g(\nabla_XY,Z)+g(X,\nabla_XZ)$

The Fundamental Theorem of Riemannian Geometry establishes that this $\nabla$ exists and it is unique, and it is called the Levi-Civita connection for the metric on .

Now, if one uses a coordinated patch in one has a set of n-coordinated vector fields $\partial_1,..,\partial_n$ meaning $\partial_i={{\partial}\over{\partial u^i}}$ being the coordinate functions. These are also dubbed holonomic derivations.

So it makes sense to speak about the derivatives $\nabla_{\partial_i}\partial_j$ and since the $\partial_i$ are tangent which generate at a point , then $\nabla_{\partial_i}\partial_j$ is also tangent, so there are $n\times n$ numbers (functions if one varies position) $\Gamma^s_{ij}$ which enters in the relation

$\displaystyle \nabla_{\partial_i}\partial_j=\sum_s\Gamma^s_{ij}\partial_s.$

These coefficients $\Gamma^s_{ij}$ are called Christoffel symbols and an easy calculation shows that

$\displaystyle \Gamma^k_{ij}={1\over 2}\sum_sg^{ks}[g_{sj,i}+g_{is,j}-g_{ij,s}]$

where $g_{ij}=g(\partial_i,\partial_j)$ , $g^{ij}$ are the entries of the matrix $[g_{ij}]^{-1}$ and $g_{ij,k}=\partial_k(g_{ij})$ .

Routinely one can check that under a change of coordinates $u^i\to w^j$ these functions transform as

$\displaystyle \bar{\Gamma}^i_{kl}= {{\partial w^i}\over{\partial u^m}}{{\partia... ...rtial}^2u^p\over{\partial w^k\partial w^l}}{{\partial w^i}\over{\partial u^p}}$

here we have used Einstein's sum convention (

-sums) and the term

$\displaystyle { {\partial}^2u^p\over{\partial w^k\partial w_l} } { {\partial w^i}\over{\partial u^p} }$

shows that the $\Gamma^i_{kl}$ are not tensors.

For a proof please see the last part in: http://planetmath.org/?op=getobj&from=collab&id=64

Connection with base vectors.

Let us assume that coordinates are referred to a right-handed orthogonal Cartesian system with attached constant base vectors $\mathbf{e}_i\equiv\mathbf{e}^i$ and coordinates referred to a general curvilinear system attached to a local covariant base vectors $\mathbf{g}_j$ and local contravariant base vectors $\mathbf{g}^k$ , both systems embedded in the Euclidean space $\mathbb{R}^n$ . We shall also suppose diffeomorphic the transfomation $u^i\mapsto w^j$ . Then, by definition

$\displaystyle \mathbf{g}_j:=\frac{\partial u^i}{\partial w^j}\mathbf{e}_i\:, \qquad \mathbf{g}^j:=\frac{\partial w^j}{\partial u^i}\mathbf{e}^i\:,$

(1)

and its inverses

$\displaystyle \mathbf{e}_i=\mathbf{e}^i=\frac{\partial u^i}{\partial w^j}\mathbf{g}^j= \frac{\partial w^j}{\partial u^i}\mathbf{g}_j\:.$

(2)

Let us consider differentiation of base vectors $\mathbf{g}_j$ , which may be written from (1),(2)

$\displaystyle \frac{\partial\mathbf{g}_j}{\partial w^k}= \frac{\partial^2 u^i}{... ...s}{\partial u^i}\mathbf{g}_s\equiv \frac{\partial\mathbf{g}_k}{\partial w^j}\:,$

and using the Christoffel symbols this becomes

$\displaystyle \frac{\partial\mathbf{g}_j}{\partial w^k}=\Gamma_{jks}\mathbf{g}^s= \Gamma^r_{jk}\mathbf{g}_r\:,$

(3)

where

$\displaystyle \Gamma_{jks}=\frac{\partial^2 u^i}{\partial w^j\partial w^k} \frac{\partial u^i}{\partial w^s}\:, \qquad \Gamma^r_{jk}=g^{rs}\Gamma_{jks}\:.$

(4)

Since the transformation of covariant and contravariant metric tensors are given by

$\displaystyle g_{jk}=\frac{\partial u^i}{\partial w^j}\frac{\partial u^l}{\part... ...rac{\partial w^j}{\partial u^i} \frac{\partial w^k}{\partial u^l}\delta^{il}\:,$

is easy to see from here that Christoffel symbol $\Gamma_{jks}$ enjoy the property

$\displaystyle \Gamma_{jks}=\frac{1}{2}\bigg(\frac{\partial g_{js}}{\partial w^k... ...artial g_{ks}}{\partial w^j}- \frac{\partial g_{jk}}{\partial w^s}\bigg)\:\cdot$

(5)

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

$\displaystyle \frac{\partial\mathbf{g}^j}{\partial w^k}=-\Gamma^j_{ks}\mathbf{g}^s\:.$

(6)

Is easy to show the following results:

$\displaystyle \Gamma_{jks}=\Gamma_{kjs}= \mathbf{g}_s\cdot\frac{\partial\mathbf... ...k}{\partial w^j}= \mathbf{g}_s\cdot\frac{\partial\mathbf{g}_j}{\partial w^k}\:,$

$\displaystyle \Gamma^r_{jk}=\Gamma^r_{kj}=\mathbf{g}^r\cdot \frac{\partial\math... ...}{\partial w^j}= -\mathbf{g}_j\cdot\frac{\partial\mathbf{g}^r}{\partial w^k}\:,$

$\displaystyle \Gamma^i_{ir}=\frac{1}{2}g^{is}(g_{is,r}+g_{rs,i}-g_{ir,s})= \fra... ...{is}}{\partial w^r}= \frac{1}{\sqrt{g}}\frac{\partial\sqrt{g}}{\partial w^r}\:,$

$\displaystyle \Gamma_{jsk}+\Gamma_{ksj}=g_{jk,s}\:,$

comma denoting differentiation with respect to the curvilinear coordinates

and $g=\vert g_{jk}\vert$ . When the coordinate curves are orthogonal we have the following formulae for the Christoffel symbols: (repeated indices are not to be summed)

$\displaystyle \Gamma_{jks}=0\:, \qquad \Gamma^s_{jk}=0\:, \qquad (j\neq k\neq s\neq j),$

$\displaystyle \Gamma_{iir}=-\frac{1}{2}\frac{\partial g_{ii}}{\partial w^r}\:, ... ...}=-\frac{1}{2g_{rr}}\frac{\partial g_{ii}}{\partial w^r}\:, \qquad (r\neq i)\:,$

$\displaystyle \Gamma_{iri}=\Gamma_{rii}=\frac{1}{2}\frac{\partial g_{ii}}{\part... ...rr}}{\partial w^i}= \frac{1}{2}\frac{\partial\log{g_{rr}}}{\partial w^i}\:\cdot$

"Christoffel symbols" is owned by juanman. [ full author list (4) ]

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