# Christoffel symbols

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

Or as a section ${\mathbb{R}}^{n}\lx@stackrel{{\scriptstyle V}}{{\to}}T({\mathbb{R}}^{n})$ 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 $p$.

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

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

 $dX|_{p}(Y(p))$

measures how $X$ varies in the direction $Y$ at $p$.

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

 $D_{Y}X$

We can be consider it as a bilinear map

 $D:T({\mathbb{R}}^{n})\times T({\mathbb{R}}^{n})\to T({\mathbb{R}}^{n}).$
 $(X,Y)\mapsto D_{X}Y$

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

1. 1.

$D_{fY}X=fD_{Y}X$

2. 2.

$D_{Y}(fX)=(Yf)X+fD_{Y}X$

3. 3.

$D_{X}Y-D_{Y}X=[X,Y]$

4. 4.

$X(Y\cdot Z)=D_{X}Y\cdot Z+X\cdot D_{X}Z$

Here we have abbreviated (as usual) $Yf=Y\cdot\nabla F$ and the operation $[X,Y]$ is the Lie bracket.

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

Now, let $M$ be a n-dimensional differentiable manifold and let $TM$ 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

 $g(X,Y)|_{p}=X(p)\cdot Y(p)$

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

It is possible construct a bilinear operator $\nabla$

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

compatible with $g$ and which satisfies the following properties

1. 1.

$\nabla_{fY}X=f\nabla_{Y}X$

2. 2.

$\nabla_{Y}(fX)=(Yf)X+f\nabla_{Y}X$

3. 3.

$\nabla_{X}Y-\nabla_{Y}X=[X,Y]$

4. 4.

$Xg(Y,Z)=g(\nabla_{X}Y,Z)+g(X,\nabla_{X}Z)$

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

Now, if one uses a coordinated patch in $M$ one has a set of n-coordinated vector fields $\partial_{1},..,\partial_{n}$ meaning $\partial_{i}={{\partial}\over{\partial u^{i}}}$ being $u^{i}$ 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 $T_{p}(M)$, 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

 $\nabla_{\partial_{i}}\partial_{j}=\sum_{s}\Gamma^{s}_{ij}\partial_{s}.$

These coefficients $\Gamma^{s}_{ij}$ are called and an easy calculation shows that

 $\Gamma^{k}_{ij}={1\over 2}\sum_{s}g^{ks}[g_{sj,i}+g_{is,j}-g_{ij,s}]$

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

 $\bar{\Gamma}^{i}_{kl}={{\partial w^{i}}\over{\partial u^{m}}}{{\partial u^{n}}% \over{\partial w^{k}}}{{\partial u^{p}}\over{\partial w^{l}}}\Gamma^{m}_{np}+{% {\partial}^{2}u^{p}\over{\partial w^{k}\partial w^{l}}}{{\partial w^{i}}\over{% \partial u^{p}}}$

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

 ${{\partial}^{2}u^{p}\over{\partial w^{k}\partial w_{l}}}{{\partial w^{i}}\over% {\partial u^{p}}}$

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=64http://planetmath.org/?op=getobj&from=collab&id=64

## Connection with base vectors.

Let us assume that coordinates $u^{i}$ are referred to a right-handed orthogonal Cartesian system with attached constant base vectors $\mathbf{e}_{i}\equiv\mathbf{e}^{i}$ and coordinates $w^{j}$ 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)

 $\displaystyle\frac{\partial\mathbf{g}_{j}}{\partial w^{k}}=\frac{\partial^{2}u% ^{i}}{\partial w^{j}\partial w^{k}}\mathbf{e}_{i}=\frac{\partial^{2}u^{i}}{% \partial w^{j}\partial w^{k}}\frac{\partial u^{i}}{\partial w^{s}}\mathbf{g}^{% s}=\frac{\partial^{2}u^{i}}{\partial w^{j}\partial w^{k}}\frac{\partial w^{s}}% {\partial u^{i}}\mathbf{g}_{s}\equiv\frac{\partial\mathbf{g}_{k}}{\partial w^{% j}}\>,$

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

 $\displaystyle g_{jk}=\frac{\partial u^{i}}{\partial w^{j}}\frac{\partial u^{l}% }{\partial w^{k}}\delta_{il}\>,\qquad g^{jk}=\frac{\partial w^{j}}{\partial u^% {i}}\frac{\partial w^{k}}{\partial u^{l}}\delta^{il}\>,$

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

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

 $\displaystyle\Gamma_{jks}=\Gamma_{kjs}=\mathbf{g}_{s}\cdot\frac{\partial% \mathbf{g}_{k}}{\partial w^{j}}=\mathbf{g}_{s}\cdot\frac{\partial\mathbf{g}_{j% }}{\partial w^{k}}\>,$
 $\displaystyle\Gamma^{r}_{jk}=\Gamma^{r}_{kj}=\mathbf{g}^{r}\cdot\frac{\partial% \mathbf{g}_{j}}{\partial w^{k}}=\mathbf{g}^{r}\cdot\frac{\partial\mathbf{g}_{k% }}{\partial w^{j}}=-\mathbf{g}_{j}\cdot\frac{\partial\mathbf{g}^{r}}{\partial w% ^{k}}\>,$
 $\displaystyle\Gamma^{i}_{ir}=\frac{1}{2}g^{is}(g_{is,r}+g_{rs,i}-g_{ir,s})=% \frac{1}{2}g^{is}g_{is,r}=\frac{1}{2g}\frac{\partial g}{\partial g_{is}}\frac{% \partial g_{is}}{\partial w^{r}}=\frac{1}{\sqrt{g}}\frac{\partial\sqrt{g}}{% \partial w^{r}}\>,$
 $\displaystyle\Gamma_{jsk}+\Gamma_{ksj}=g_{jk,s}\>,$

comma denoting differentiation with respect to the curvilinear coordinates $w^{j}$ and $g=|g_{jk}|$. When the coordinate curves are orthogonal we have the following formulae for the Christoffel symbols: (repeated indices are not to be summed)

 $\displaystyle\Gamma_{jks}=0\>,\qquad\Gamma^{s}_{jk}=0\>,\qquad(j\neq k\neq s% \neq j),$
 $\displaystyle\Gamma_{iir}=-\frac{1}{2}\frac{\partial g_{ii}}{\partial w^{r}}\>% ,\qquad\Gamma^{r}_{ii}=-\frac{1}{2g_{rr}}\frac{\partial g_{ii}}{\partial w^{r}% }\>,\qquad(r\neq i)\>,$
 $\displaystyle\Gamma_{iri}=\Gamma_{rii}=\frac{1}{2}\frac{\partial g_{ii}}{% \partial w^{r}}\>,\qquad\Gamma^{r}_{ri}=\Gamma^{r}_{ir}=\frac{1}{2g_{rr}}\frac% {\partial g_{rr}}{\partial w^{i}}=\frac{1}{2}\frac{\partial\log{g_{rr}}}{% \partial w^{i}}\>\cdot$
Title Christoffel symbols ChristoffelSymbols 2013-03-22 15:43:52 2013-03-22 15:43:52 juanman (12619) juanman (12619) 24 juanman (12619) Definition msc 53B20 msc 53-01 connection coefficients Connection