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Christoffel Symbol of the Second Kind

Variously denoted $\left\{{m\atop i\quad j}\right\}$ or $\Gamma^m_{ij}$.

$\displaystyle \left\{{m\atop i\quad j}\right\}$ $\textstyle \equiv$ $\displaystyle \Gamma^m_{ij} = \vec e\,^m \cdot {\partial \vec e_i\over \partial q^j} = g^{km}[ij,k]$  
  $\textstyle =$ $\displaystyle {1\over 2}g^{km}\left({{\partial g_{ik}\over\partial q^j} +{\partial g_{jk}\over \partial q^i} -{\partial g_{ij}\over \partial q^k}}\right),$ (1)

where $\Gamma^m_{ij}$ is a Connection Coefficient and $\{bc, d\}$ is a Christoffel Symbol of the First Kind.
\left\{{a\atop b\quad c}\right\} = g_{ad}\{bc,d\}.
\end{displaymath} (2)

The Christoffel symbols are given in terms of the first Fundamental Form $E$, $F$, and $G$ by
$\displaystyle \Gamma_{11}^1$ $\textstyle =$ $\displaystyle {GE_u-2FF_u+FE_v\over 2(EG-F^2)}$ (3)
$\displaystyle \Gamma_{12}^1$ $\textstyle =$ $\displaystyle {GE_v-FG_u\over 2(EG-F^2)}$ (4)
$\displaystyle \Gamma_{22}^1$ $\textstyle =$ $\displaystyle {2GF_v-GG_u-FG_v\over 2(EG-F^2)}$ (5)
$\displaystyle \Gamma_{11}^2$ $\textstyle =$ $\displaystyle {2EF_u-EE_v-FE_u\over 2(EG-F^2)}$ (6)
$\displaystyle \Gamma_{12}^2$ $\textstyle =$ $\displaystyle {EG_u-FE_v\over 2(EG-F^2)}$ (7)
$\displaystyle \Gamma_{22}^2$ $\textstyle =$ $\displaystyle {EG_v-2FF_v+FG_u\over 2(EG-F^2)},$ (8)

and $\Gamma_{21}^1=\Gamma_{12}^1$ and $\Gamma_{21}^2=\Gamma_{12}^2$. If $F=0$, the Christoffel symbols of the second kind simplify to
$\displaystyle \Gamma_{11}^1$ $\textstyle =$ $\displaystyle {E_u\over 2E}$ (9)
$\displaystyle \Gamma_{12}^1$ $\textstyle =$ $\displaystyle {E_v\over 2E}$ (10)
$\displaystyle \Gamma_{22}^1$ $\textstyle =$ $\displaystyle -{G_u\over 2E}$ (11)
$\displaystyle \Gamma_{11}^2$ $\textstyle =$ $\displaystyle -{E_v\over 2G}$ (12)
$\displaystyle \Gamma_{12}^2$ $\textstyle =$ $\displaystyle {G_u\over 2G}$ (13)
$\displaystyle \Gamma_{22}^2$ $\textstyle =$ $\displaystyle {G_v\over 2G}$ (14)

(Gray 1993).

The following relationships hold between the Christoffel symbols of the second kind and coefficients of the first Fundamental Form,

$\displaystyle \Gamma_{11}^1 E+\Gamma_{11}^2 F$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}E_u$ (15)
$\displaystyle \Gamma_{12}^1 E+\Gamma_{12}^2 F$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}E_v$ (16)
$\displaystyle \Gamma_{22}^1 E+\Gamma_{22}^2 F$ $\textstyle =$ $\displaystyle F_v-{\textstyle{1\over 2}}G_u$ (17)
$\displaystyle \Gamma_{11}^1 F+\Gamma_{11}^2 G$ $\textstyle =$ $\displaystyle F_u-{\textstyle{1\over 2}}E_v$ (18)
$\displaystyle \Gamma_{12}^1 F+\Gamma_{12}^2 G$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}G_u$ (19)
$\displaystyle \Gamma_{22}^1 F+\Gamma_{22}^2 G$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}G_v$ (20)
$\displaystyle \Gamma_{11}^1+\Gamma_{12}^2$ $\textstyle =$ $\displaystyle (\ln\sqrt{EG-F^2}\,)_u$ (21)
$\displaystyle \Gamma_{12}^1+\Gamma_{22}^2$ $\textstyle =$ $\displaystyle (\ln\sqrt{EG-F^2}\,)_v$ (22)

(Gray 1993).

For a surface given in Monge's Form $z=F(x,y)$,

\Gamma^k_{ij}={z_{ij}z_k\over 1+{z_1}^2+{z_2}^2}.
\end{displaymath} (23)

See also Christoffel Symbol of the First Kind, Connection Coefficient, Gauss Equations


Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 160-167, 1985.

Gray, A. ``Christoffel Symbols.'' §20.3 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 397-400, 1993.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 47-48, 1953.

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© 1996-9 Eric W. Weisstein