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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 $ p$.

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

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}}$
  1. $ D_{fY}X=fD_YX$
  2. $ D_Y(fX)=(Yf)X+ fD_YX$
  3. $ D_XY-D_YX=[X,Y]$
  4. $ 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 $ [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

$\displaystyle g(X,Y)\vert _p=X(p)\cdot Y(p)$
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 $ g$ and which satisfies the following properties
  1. $ \nabla_{fY}X=f\nabla_YX$
  2. $ \nabla_Y(fX)=(Yf)X+ f\nabla_YX$
  3. $ \nabla_XY-\nabla_YX=[X,Y]$
  4. $ 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 $ 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

$\displaystyle \nabla_{\partial_i}\partial_j=\sum_s\Gamma^s_{ij}\partial_s.$
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 ($ m,n,p$-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 $ 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)
$\displaystyle \frac{\partial\mathbf{g}_j}{\partial w^k}= \frac{\partial^2 u^i}{... ...s}{\partial u^i}\mathbf{g}_s\equiv \frac{\partial\mathbf{g}_k}{\partial w^j}\:,$    

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 $ w^j$ 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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See Also: connection

Other names:  connection coefficients
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Cross-references: curves, similar, metric tensors, transformation, differentiation, inverses, diffeomorphic, Euclidean space, vectors, base, orthogonal, right-handed, proof, tensors, term, sum, Transform, change of coordinates, matrix, coefficients, relation, numbers, generate, tangent, derivatives, coordinate, metric, Levi-Civita connection, geometry, properties, compatible, inner product, Lie algebra, differentiable manifold, connection, Lie bracket, operation, bilinear map, obvious, components, measures, point, Jacobian matrix, scalars, derivations, linear operators, tangent space, tangent bundle, section, map, differentiable, vector field
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This is version 21 of Christoffel symbols, born on 2006-03-04, modified 2006-08-02.
Object id is 7681, canonical name is Christoffel20symbols.
Accessed 5747 times total.

Classification:
AMS MSC53-01 (Differential geometry :: Instructional exposition )
 53B20 (Differential geometry :: Local differential geometry :: Local Riemannian geometry)

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