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证明：当$(u,v)$是曲面的正交曲率线网时 ,Codazzi方程可以简化为$L_v=H{E}_v,N_u=H{G}_u$

证：

回忆Codazzi方程：

$$\{\begin{array}{l}\frac{\partial b_{11}}{\partial u^2}-\frac{\partial b_{12}}{\partial u^1}=-b_{2\delta }{\Gamma }_{11}^{\delta }+b_{1\delta }{\Gamma }_{12}^{\delta }\\ \frac{\partial b_{21}}{\partial u^2}-\frac{\partial b_{22}}{\partial u^1}=-b_{2\delta }{\Gamma }_{21}^{\delta }+b_{1\delta }{\Gamma }_{22}^{\delta }\end{array}$$

由于是正交曲率线网，$F=M=0$。也就是$g_{12}=b_{12}=0$ 。先代入$b_{12}=0$,得到

$$\{\begin{array}{l}\frac{\partial b_{11}}{\partial u^2}=-b_{22}{\Gamma }_{11}^2+b_{11}{\Gamma }_{12}^1\\ -\frac{\partial b_{22}}{\partial u^1}=-b_{22}{\Gamma }_{21}^2+b_{11}{\Gamma }_{22}^1\end{array}$$

并回忆正交参数系下的Christoffel符号：

$$\begin{gathered} {\Gamma }_{11}^1=\frac{1}{2}(\frac{\ln \partial E}{\partial u^1}),{\Gamma }_{11}^2=-\frac{1}{2G}(\frac{\partial E}{\partial u^2}) \\ {\Gamma }_{22}^1=-\frac{1}{2E}(\frac{\partial G}{\partial u^1}),{\Gamma }_{22}^2=\frac{1}{2}(\frac{\partial \ln G}{\partial u^2}) \\ {\Gamma }_{12}^1=\frac{1}{2}(\frac{\partial \ln E}{\partial u^2}),{\Gamma }_{12}^2=\frac{1}{2}(\frac{\partial \ln G}{\partial u^1}) \end{gathered}$$

分别计算

$$\begin{gathered} -b_{22}{\Gamma }_{11}^2+b_{11}{\Gamma }_{12}^1=N\frac{1}{2G}(\frac{\partial E}{\partial u^2})+L\frac{1}{2E}(\frac{\partial E}{\partial u^2}) \\ -b_{22}{\Gamma }_{21}^2+b_{11}{\Gamma }_{22}^1=-N\frac{1}{2G}(\frac{\partial G}{\partial u^1})-L\frac{1}{2E}(\frac{\partial G}{\partial u^1}) \end{gathered}$$

正交参数系下的平均曲率是

$$H=\frac{LG+NE}{2EG}$$

这就得到$L_v=H{E}_v,N_u=H{G}_u$ 。