正交曲率线网时的Codazzi方程
证明：当 ( u , v ) (u,v) (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 L_v=HE_v,N_u=HG_u Lv=HEv,Nu=HGu
证：
回忆Codazzi方程：
{ ∂ b 11 ∂ u 2 − ∂ b 12 ∂ u 1 = − b 2 δ Γ 11 δ + b 1 δ Γ 12 δ ∂ b 21 ∂ u 2 − ∂ b 22 ∂ u 1 = − b 2 δ Γ 21 δ + b 1 δ Γ 22 δ \left.\left\{\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}\right.\right. {∂u2∂b11−∂u1∂b12=−b2δΓ11δ+b1δΓ12δ∂u2∂b21−∂u1∂b22=−b2δΓ21δ+b1δΓ22δ
由于是正交曲率线网， F = M = 0 F=M=0 F=M=0。也就是 g 12 = b 12 = 0 g_{12}=b_{12}=0 g12=b12=0 。先代入 b 12 = 0 b_{12}=0 b12=0,得到
{ ∂ b 11 ∂ u 2 = − b 22 Γ 11 2 + b 11 Γ 12 1 − ∂ b 22 ∂ u 1 = − b 22 Γ 21 2 + b 11 Γ 22 1 \left.\begin{cases}\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{cases}\right. {∂u2∂b11=−b22Γ112+b11Γ121−∂u1∂b22=−b22Γ212+b11Γ221
并回忆正交参数系下的Christoffel符号：
Γ 11 1 = 1 2 ( ln ∂ E ∂ u 1 ) , Γ 11 2 = − 1 2 G ( ∂ E ∂ u 2 ) Γ 22 1 = − 1 2 E ( ∂ G ∂ u 1 ) , Γ 22 2 = 1 2 ( ∂ ln G ∂ u 2 ) Γ 12 1 = 1 2 ( ∂ ln E ∂ u 2 ) , Γ 12 2 = 1 2 ( ∂ ln G ∂ u 1 ) \Gamma_{11}^1=\frac12(\frac{\ln\partial E}{\partial u^1}),\Gamma_{11}^2=-\frac1{2G}(\frac{\partial E}{\partial u^2})\\\Gamma_{22}^1=-\frac1{2E}(\frac{\partial G}{\partial u^1}),\Gamma_{22}^2=\frac12(\frac{\partial\ln G}{\partial u^2})\\\Gamma_{12}^1=\frac12(\frac{\partial\ln E}{\partial u^2}),\Gamma_{12}^2=\frac12(\frac{\partial\ln G}{\partial u^1}) Γ111=21(∂u1ln∂E),Γ112=−2G1(∂u2∂E)Γ221=−2E1(∂u1∂G),Γ222=21(∂u2∂lnG)Γ121=21(∂u2∂lnE),Γ122=21(∂u1∂lnG)
分别计算
− b 22 Γ 11 2 + b 11 Γ 12 1 = N 1 2 G ( ∂ E ∂ u 2 ) + L 1 2 E ( ∂ E ∂ u 2 ) − b 22 Γ 21 2 + b 11 Γ 22 1 = − N 1 2 G ( ∂ G ∂ u 1 ) − L 1 2 E ( ∂ G ∂ u 1 ) -b_{22}\Gamma_{11}^2+b_{11}\Gamma_{12}^1=N\frac1{2G}(\frac{\partial E}{\partial u^2})+L\frac1{2E}(\frac{\partial E}{\partial u^2})\\-b_{22}\Gamma_{21}^2+b_{11}\Gamma_{22}^1=-N\frac1{2G}(\frac{\partial G}{\partial u^1})-L\frac1{2E}(\frac{\partial G}{\partial u^1}) −b22Γ112+b11Γ121=N2G1(∂u2∂E)+L2E1(∂u2∂E)−b22Γ212+b11Γ221=−N2G1(∂u1∂G)−L2E1(∂u1∂G)
正交参数系下的平均曲率是
H = L G + N E 2 E G H=\frac{LG+NE}{2EG} H=2EGLG+NE
这就得到 L v = H E v , N u = H G u L_v=HE_v,N_u=HG_u Lv=HEv,Nu=HGu 。
原文地址：https://blog.csdn.net/weixin_73404807/article/details/144328687
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