Christoffel symbols

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

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

Another viewpoint about tangent vectors is that they are also linear operators called derivations and they actover scalars $f:{ℝ}^n\to ℝ$ 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 $p\in {ℝ}^n$.Then, for any other vector field $Y:{ℝ}^n\to {ℝ}^n$,

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

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

We have ${dX|}_p(Y(p))=({Y(p)\cdot \nabla X^1|}_p,\mathrm{\dots },{Y(p)\cdot \nabla X^n|}_p)$, 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 ${ℝ}^n$ which is symbolized as

$$D_{Y}X$$

We can be consider it as a bilinear map

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

$$(X,Y)\mapsto D_{X}Y$$

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

1. $D_{fY}X=f{D}_{Y}X$

2. $D_Y(fX)=(Yf)X+f{D}_{Y}X$

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

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 ${ℝ}^n$.

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

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

in each $T_p(M)\equiv {ℝ}^n$.

It is possible construct a bilinear operator $\nabla$

$$\nabla :\mathrm{\Gamma }(M)\times \mathrm{\Gamma }(M)\to \mathrm{\Gamma }(M)$$

compatible with $g$ and which satisfies the following properties

1. ${\nabla }_{fY}X=f{\nabla }_{Y}X$

2. ${\nabla }_Y(fX)=(Yf)X+f{\nabla }_{Y}X$

3. ${\nabla }_{X}Y-{\nabla }_{Y}X=[X,Y]$

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 Levi-Civita connection 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 ${\partial }_i=\frac{\partial }{\partial u^i}$ 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) ${\mathrm{\Gamma }}_{ij}^s$ which entersin the relation

$${\nabla }_{{\partial }_i}{\partial }_j=\sum _s{\mathrm{\Gamma }}_{ij}^s{\partial }_s.$$

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

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

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

$${\overline{\mathrm{\Gamma }}}_{kl}^i=\frac{\partial w^i}{\partial u^m}\frac{\partial u^n}{\partial w^k}\frac{\partial u^p}{\partial w^l}{\mathrm{\Gamma }}_{np}^m+\frac{{\partial }^2{u}^p}{\partial w^k\partial w^l}\frac{\partial w^i}{\partial u^p}$$

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

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

shows that the ${\mathrm{\Gamma }}_{kl}^i$ 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 ${𝐞}_i\equiv {𝐞}^i$ and coordinates $w^j$ referred to a general curvilinear system attached to a local covariant base vectors ${𝐠}_j$ and local contravariant base vectors ${𝐠}^k$, both systems embedded in the Euclidean space ${ℝ}^n$. We shall also suppose diffeomorphic the transfomation $u^i\mapsto w^j$. Then, by definition

${𝐠}_j:={\displaystyle \frac{\partial u^i}{\partial w^j}}{𝐞}_i,{𝐠}^j:={\displaystyle \frac{\partial w^j}{\partial u^i}}{𝐞}^i,$(1)

and its inverses

${𝐞}_i={𝐞}^i={\displaystyle \frac{\partial u^i}{\partial w^j}}{𝐠}^j={\displaystyle \frac{\partial w^j}{\partial u^i}}{𝐠}_j.$(2)

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

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

and using the Christoffel symbols this becomes

${\displaystyle \frac{\partial {𝐠}_j}{\partial w^k}}={\mathrm{\Gamma }}_{jks}{𝐠}^s={\mathrm{\Gamma }}_{jk}^r{𝐠}_r,$(3)

where

${\mathrm{\Gamma }}_{jks}={\displaystyle \frac{{\partial }^2{u}^i}{\partial w^j\partial w^k}}{\displaystyle \frac{\partial u^i}{\partial w^s}},{\mathrm{\Gamma }}_{jk}^r=g^{rs}{\mathrm{\Gamma }}_{jks}.$(4)

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

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

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

${\mathrm{\Gamma }}_{jks}={\displaystyle \frac{1}{2}}({\displaystyle \frac{\partial g_{js}}{\partial w^k}}+{\displaystyle \frac{\partial g_{ks}}{\partial w^j}}-{\displaystyle \frac{\partial g_{jk}}{\partial w^s}})\cdot$(5)

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

${\displaystyle \frac{\partial {𝐠}^j}{\partial w^k}}=-{\mathrm{\Gamma }}_{ks}^j{𝐠}^s.$(6)

Is easy to show the following results:

${\mathrm{\Gamma }}_{jks}={\mathrm{\Gamma }}_{kjs}={𝐠}_s\cdot {\displaystyle \frac{\partial {𝐠}_k}{\partial w^j}}={𝐠}_s\cdot {\displaystyle \frac{\partial {𝐠}_j}{\partial w^k}},$

${\mathrm{\Gamma }}_{jk}^r={\mathrm{\Gamma }}_{kj}^r={𝐠}^r\cdot {\displaystyle \frac{\partial {𝐠}_j}{\partial w^k}}={𝐠}^r\cdot {\displaystyle \frac{\partial {𝐠}_k}{\partial w^j}}=-{𝐠}_j\cdot {\displaystyle \frac{\partial {𝐠}^r}{\partial w^k}},$

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

${\mathrm{\Gamma }}_{jsk}+{\mathrm{\Gamma }}_{ksj}=g_{jk,s},$

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)

${\mathrm{\Gamma }}_{jks}=0,{\mathrm{\Gamma }}_{jk}^s=0,(j\ne k\ne s\ne j),$

${\mathrm{\Gamma }}_{iir}=-{\displaystyle \frac{1}{2}}{\displaystyle \frac{\partial g_{ii}}{\partial w^r}},{\mathrm{\Gamma }}_{ii}^r=-{\displaystyle \frac{1}{2g_{rr}}}{\displaystyle \frac{\partial g_{ii}}{\partial w^r}},(r\ne i),$

${\mathrm{\Gamma }}_{iri}={\mathrm{\Gamma }}_{rii}={\displaystyle \frac{1}{2}}{\displaystyle \frac{\partial g_{ii}}{\partial w^r}},{\mathrm{\Gamma }}_{ri}^r={\mathrm{\Gamma }}_{ir}^r={\displaystyle \frac{1}{2g_{rr}}}{\displaystyle \frac{\partial g_{rr}}{\partial w^i}}={\displaystyle \frac{1}{2}}{\displaystyle \frac{\partial \log g_{rr}}{\partial w^i}}\cdot$