Christoffel Symbol of the Second Kind

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

$$\left\{{m\atop i\quad j}\right\} \equiv \Gamma^m_{ij} = \vec e\,^m \cdot {\partial \vec e_i\over \partial q^j} = g^{km}[ij,k]$$

$$= {1\over 2}g^{km}\left({{\partial g_{ik}\over\partial q^j} +{\partial g_{jk}\over \partial q^i} -{\partial g_{ij}\over \partial q^k}}\right),$$ (1)

where $\Gamma^m_{ij}$ is a Connection Coefficient and $\{bc, d\}$ is a Christoffel Symbol of the First Kind.

$$\left\{{a\atop b\quad c}\right\} = g_{ad}\{bc,d\}.$$ (2)

The Christoffel symbols are given in terms of the first Fundamental Form $E$, $F$, and $G$ by

$$\Gamma_{11}^1 = {GE_u-2FF_u+FE_v\over 2(EG-F^2)}$$ (3)

$$\Gamma_{12}^1 = {GE_v-FG_u\over 2(EG-F^2)}$$ (4)

$$\Gamma_{22}^1 = {2GF_v-GG_u-FG_v\over 2(EG-F^2)}$$ (5)

$$\Gamma_{11}^2 = {2EF_u-EE_v-FE_u\over 2(EG-F^2)}$$ (6)

$$\Gamma_{12}^2 = {EG_u-FE_v\over 2(EG-F^2)}$$ (7)

$$\Gamma_{22}^2 = {EG_v-2FF_v+FG_u\over 2(EG-F^2)},$$ (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

$$\Gamma_{11}^1 = {E_u\over 2E}$$ (9)

$$\Gamma_{12}^1 = {E_v\over 2E}$$ (10)

$$\Gamma_{22}^1 = -{G_u\over 2E}$$ (11)

$$\Gamma_{11}^2 = -{E_v\over 2G}$$ (12)

$$\Gamma_{12}^2 = {G_u\over 2G}$$ (13)

$$\Gamma_{22}^2 = {G_v\over 2G}$$ (14)

(Gray 1993).

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

$$\Gamma_{11}^1 E+\Gamma_{11}^2 F = {\textstyle{1\over 2}}E_u$$ (15)

$$\Gamma_{12}^1 E+\Gamma_{12}^2 F = {\textstyle{1\over 2}}E_v$$ (16)

$$\Gamma_{22}^1 E+\Gamma_{22}^2 F = F_v-{\textstyle{1\over 2}}G_u$$ (17)

$$\Gamma_{11}^1 F+\Gamma_{11}^2 G = F_u-{\textstyle{1\over 2}}E_v$$ (18)

$$\Gamma_{12}^1 F+\Gamma_{12}^2 G = {\textstyle{1\over 2}}G_u$$ (19)

$$\Gamma_{22}^1 F+\Gamma_{22}^2 G = {\textstyle{1\over 2}}G_v$$ (20)

$$\Gamma_{11}^1+\Gamma_{12}^2 = (\ln\sqrt{EG-F^2}\,)_u$$ (21)

$$\Gamma_{12}^1+\Gamma_{22}^2 = (\ln\sqrt{EG-F^2}\,)_v$$ (22)

(Gray 1993).

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

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

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

References

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.