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''of a differential quadratic form
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[[Category:TeX done]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c0222201.png" /></td> </tr></table>
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''of a differential quadratic form''
  
''
+
\begin{equation*}
 +
\sum_{r,s=1}^ng_{rs}dx^rdx^s.
 +
\end{equation*}
  
 
An abbreviated notation for the expression
 
An abbreviated notation for the expression
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c0222202.png" /></td> </tr></table>
+
\begin{equation*}
 +
\frac{1}{2}\left(\frac{\partial g_{ij}}{\partial x^j}+\frac{\partial g_{jk}}{\partial x^i}-\frac{\partial g_{ij}}{\partial x^k}\right)\equiv \Gamma_{k,ij}.
 +
\end{equation*}
  
The symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c0222203.png" /> are called the Christoffel symbols of the first kind, in contrast to the Christoffel symbols of the second kind, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c0222204.png" />, defined by
+
The symbols $\Gamma_{k,ij}$ are called the Christoffel symbols of the first kind, in contrast to the Christoffel symbols of the second kind, $\Gamma^k_{ij}$, defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c0222205.png" /></td> </tr></table>
+
\begin{equation*}
 +
\Gamma^k_{ij}=\sum_{t=1}^ng^{kt}\Gamma_{t,ij},
 +
\end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c0222206.png" /> is defined as follows:
+
where $g^{kt}$ is defined as follows:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c0222207.png" /></td> </tr></table>
+
\begin{equation*}
 +
\sum_{k=1}^ng^{kt}g_{ks}=
 +
\begin{cases}
 +
1\qquad\text{if }t=s,\\
 +
0\qquad\text{if }t\neq s.
 +
\end{cases}
 +
\end{equation*}
  
 
These symbols were introduced by E.B. Christoffel in 1869.
 
These symbols were introduced by E.B. Christoffel in 1869.
  
 
====Comments====
 
====Comments====
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c0222208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c0222209.png" />, be a [[Linear connection|linear connection]] on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222011.png" /> denotes the space of vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222012.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222013.png" /> be a chart of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222014.png" />. Then on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222016.png" /> is completely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222018.png" /> are coordinates on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222019.png" />. The Christoffel symbols of the connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222020.png" /> are now given by
+
Let $\nabla:V(M)\times V(M)\rightarrow V(M)$, $(X,Y)\mapsto\nabla_XY$, be a [[Linear connection|linear connection]] on a manifold $M$, where $V(M)$ denotes the space of vector fields on $M$. Let $(U,\phi)$ be a chart of $M$. Then on $U$, $\nabla$ is completely determined by $\nabla_{\partial/\partial x^i}(\partial/\partial x^j)$, where $x^1,\dots,x^n$ are coordinates on $U$. The Christoffel symbols of the connection $\nabla$ are now given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222021.png" /></td> </tr></table>
+
\begin{equation*}
 +
\nabla_{\partial/\partial x_i}(\frac{\partial}{\partial x^j})=\sum_k\Gamma^k_{ij}\frac{\partial}{\partial x^k}.
 +
\end{equation*}
  
It is important to note that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222022.png" /> are not the components of a tensor field. In fact if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222023.png" /> denote the Christoffel symbols of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222024.png" /> with respect to a second set of coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222025.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222026.png" />, then
+
It is important to note that the $\Gamma^k_{ij}$ are not the components of a tensor field. In fact if the $\bar{\Gamma}^k_{ij}$ denote the Christoffel symbols of $\nabla$ with respect to a second set of coordinates $\bar{x}^1,\dots,\bar{x}^n$ on $U$, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222027.png" /></td> </tr></table>
+
\begin{equation*}
 +
\bar{\Gamma}^k_{ij}=\sum_{a,b,c}\Gamma^c_{ab}\frac{\partial x^a}{\partial \bar{x}^i}\frac{\partial x^b}{\partial\bar{x}^j}\frac{\partial \bar{x}^k}{\partial x^c}+\sum_c\frac{\partial^2x^c}{\partial\bar{x}^i\partial\bar{x}^j}\frac{\partial\bar{x}^k}{\partial x^c}.
 +
\end{equation*}
  
Let now <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222028.png" /> be the Riemannian connection (cf. [[Riemannian geometry|Riemannian geometry]]) defined by a (local) Riemannian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222029.png" />. Then the Christoffel symbols of this quadratic differential form are those of the connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222030.png" />. I.e.,
+
Let now $\nabla$ be the Riemannian connection (cf. [[Riemannian geometry|Riemannian geometry]]) defined by a (local) Riemannian metric $\sum_{r,s}g_{rs}dx^rdx^s$. Then the Christoffel symbols of this quadratic differential form are those of the connection $\nabla$. I.e.,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222031.png" /></td> </tr></table>
+
\begin{equation*}
 +
\langle\nabla_{\partial/\partial x^k}(\frac{\partial}{\partial x^j}),\frac{\partial}{\partial x^i}\rangle=\Gamma_{i,kj}=\frac{1}{2}\left(\frac{\partial g_{ij}}{\partial x^j}+\frac{\partial g_{jk}}{\partial x^i}-\frac{\partial g_{ij}}{\partial x^k}\right),
 +
\end{equation*}
  
 
so that indeed
 
so that indeed
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222032.png" /></td> </tr></table>
+
\begin{equation*}
 +
\nabla_{\partial/\partial x_k}(\frac{\partial}{\partial x^j})=\sum_i\Gamma^i_{kj}\frac{\partial}{\partial x^i}
 +
\end{equation*}
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022220/c02222033.png" /> are the Christoffel symbols of the second kind of the quadratic differential form as defined above.
+
where the $\Gamma^i_{kj}$ are the Christoffel symbols of the second kind of the quadratic differential form as defined above.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1''' , Interscience  (1963)  pp. Chapt. 4</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.S. Millman,  G.D. Parker,  "Elements of differential geometry" , Prentice-Hall  (1977)  pp. Chapt. 7</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,  K. Nomizu,  "Foundations of differential geometry" , '''1''' , Interscience  (1963)  pp. Chapt. 4</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.S. Millman,  G.D. Parker,  "Elements of differential geometry" , Prentice-Hall  (1977)  pp. Chapt. 7</TD></TR></table>

Latest revision as of 10:36, 23 May 2017


of a differential quadratic form

\begin{equation*} \sum_{r,s=1}^ng_{rs}dx^rdx^s. \end{equation*}

An abbreviated notation for the expression

\begin{equation*} \frac{1}{2}\left(\frac{\partial g_{ij}}{\partial x^j}+\frac{\partial g_{jk}}{\partial x^i}-\frac{\partial g_{ij}}{\partial x^k}\right)\equiv \Gamma_{k,ij}. \end{equation*}

The symbols $\Gamma_{k,ij}$ are called the Christoffel symbols of the first kind, in contrast to the Christoffel symbols of the second kind, $\Gamma^k_{ij}$, defined by

\begin{equation*} \Gamma^k_{ij}=\sum_{t=1}^ng^{kt}\Gamma_{t,ij}, \end{equation*}

where $g^{kt}$ is defined as follows:

\begin{equation*} \sum_{k=1}^ng^{kt}g_{ks}= \begin{cases} 1\qquad\text{if }t=s,\\ 0\qquad\text{if }t\neq s. \end{cases} \end{equation*}

These symbols were introduced by E.B. Christoffel in 1869.

Comments

Let $\nabla:V(M)\times V(M)\rightarrow V(M)$, $(X,Y)\mapsto\nabla_XY$, be a linear connection on a manifold $M$, where $V(M)$ denotes the space of vector fields on $M$. Let $(U,\phi)$ be a chart of $M$. Then on $U$, $\nabla$ is completely determined by $\nabla_{\partial/\partial x^i}(\partial/\partial x^j)$, where $x^1,\dots,x^n$ are coordinates on $U$. The Christoffel symbols of the connection $\nabla$ are now given by

\begin{equation*} \nabla_{\partial/\partial x_i}(\frac{\partial}{\partial x^j})=\sum_k\Gamma^k_{ij}\frac{\partial}{\partial x^k}. \end{equation*}

It is important to note that the $\Gamma^k_{ij}$ are not the components of a tensor field. In fact if the $\bar{\Gamma}^k_{ij}$ denote the Christoffel symbols of $\nabla$ with respect to a second set of coordinates $\bar{x}^1,\dots,\bar{x}^n$ on $U$, then

\begin{equation*} \bar{\Gamma}^k_{ij}=\sum_{a,b,c}\Gamma^c_{ab}\frac{\partial x^a}{\partial \bar{x}^i}\frac{\partial x^b}{\partial\bar{x}^j}\frac{\partial \bar{x}^k}{\partial x^c}+\sum_c\frac{\partial^2x^c}{\partial\bar{x}^i\partial\bar{x}^j}\frac{\partial\bar{x}^k}{\partial x^c}. \end{equation*}

Let now $\nabla$ be the Riemannian connection (cf. Riemannian geometry) defined by a (local) Riemannian metric $\sum_{r,s}g_{rs}dx^rdx^s$. Then the Christoffel symbols of this quadratic differential form are those of the connection $\nabla$. I.e.,

\begin{equation*} \langle\nabla_{\partial/\partial x^k}(\frac{\partial}{\partial x^j}),\frac{\partial}{\partial x^i}\rangle=\Gamma_{i,kj}=\frac{1}{2}\left(\frac{\partial g_{ij}}{\partial x^j}+\frac{\partial g_{jk}}{\partial x^i}-\frac{\partial g_{ij}}{\partial x^k}\right), \end{equation*}

so that indeed

\begin{equation*} \nabla_{\partial/\partial x_k}(\frac{\partial}{\partial x^j})=\sum_i\Gamma^i_{kj}\frac{\partial}{\partial x^i} \end{equation*}

where the $\Gamma^i_{kj}$ are the Christoffel symbols of the second kind of the quadratic differential form as defined above.

References

[a1] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963) pp. Chapt. 4
[a2] R.S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-Hall (1977) pp. Chapt. 7
How to Cite This Entry:
Christoffel symbol. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Christoffel_symbol&oldid=18161