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Mainardi-Codazzi Equations

$\displaystyle {\partial e\over\partial v}-{\partial f\over\partial u}$ $\textstyle =$ $\displaystyle e\Gamma_{12}^1+f(\Gamma_{12}^2-\Gamma_{11}^1)-g\Gamma_{11}^2$ (1)
$\displaystyle {\partial f\over\partial v}-{\partial g\over\partial u}$ $\textstyle =$ $\displaystyle e\Gamma_{22}^1+f(\Gamma_{22}^2-\Gamma_{12}^1)-g\Gamma_{12}^2,$ (2)

where $e$, $f$, and $g$ are coefficients of the second Fundamental Form and $\Gamma_{ij}^k$ are Christoffel Symbols of the Second Kind. Therefore,
$\displaystyle {\partial e\over\partial v}$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}E_v\left({{e\over E}+{g\over G}}\right)$ (3)
$\displaystyle {\partial g\over\partial u}$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}G_u\left({{e\over E}+{g\over G}}\right)$ (4)

$\displaystyle {\partial(\ln f)\over\partial u}$ $\textstyle =$ $\displaystyle \Gamma_{11}^1-\Gamma_{12}^2$ (5)
$\displaystyle {\partial(\ln f)\over\partial v}$ $\textstyle =$ $\displaystyle \Gamma_{22}^2-\Gamma_{12}^1$ (6)

$\displaystyle {\partial\over\partial u}\left({\ln f\over\sqrt{EG-F^2}}\right)$ $\textstyle =$ $\displaystyle -2\Gamma_{12}^2$ (7)
$\displaystyle {\partial\over\partial v}\left({\ln f\over\sqrt{EG-F^2}}\right)$ $\textstyle =$ $\displaystyle -2\Gamma_{12}^1,$ (8)

where $E$, $F$, and $G$ are coefficients of the first Fundamental Form.


Gray, A. ``The Mainardi-Codazzi Equations.'' §20.4 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 401-402, 1993.

© 1996-9 Eric W. Weisstein