Christoffel symbol

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