设 `S: bm r(u,v)` 是 `E^3` 的曲面. 称 `bm x(u,v)` 是 `S` 上的光滑向量场, 如果对任意一点 `bm r(u, v) in S`, `bm x(u, v)` 是从该点出发的一个向量, 并且 `bm x(u,v)` 光滑地依赖于参数 `(u,v)`. 对任意 `(u, v)`, 当 `bm x(u, v)` 是 `S` 在 `bm r(u, v)` 的切向量时, 称为切向量场; 当 `bm x(u, v)` 是 `S` 在 `bm r(u,v)` 的法向量时, 称为法向量场. 例如, `bm r_u, bm r_v` 是切向量场, `bm n` 是法向量场 (视它们为 `(u,v)` 的函数).
称坐标系 `{bm r(u,v)";" bm x_1(u,v), bm x_2(u,v), bm x_3(u,v)}`, `(bm x_1, bm x_2, bm x_3) != bb 0`, `AA (u,v) in D` 为曲面 `S` 上的活动标架 (Movable Frame). 一般要求 `(x_1, x_2, x_3) gt 0` 以保证这组标架为正定向的. 特别当 `{bm x_1, bm x_2, bm x_3}` 为单位正交标架时, 称为曲面 `S` 的正交活动标架, 简称正交标架. `{bm r(u,v)";" bm r_u, bm r_v, bm n}` 是活动标架的一个例子, 称为自然标架.
在这些记号下, `(g_(alpha beta)) = [E,F; F,G]`, `(b_(alpha beta)) =
[L,M; M,N]`, 且 `g_(alpha beta) = g_(beta alpha)`, `b_(alpha beta)
= b_(beta alpha)`. 我们又记
`(g^(alpha beta)) = (g_(alpha beta))^-1`,
`b_alpha^beta = b_(alpha gamma) g^(gamma beta)`,
`(delta_alpha^beta) = bm E`.
即 `g^(alpha beta)` 是 `(g_(alpha beta))` 的逆矩阵相应位置的元素,
`(b_alpha^beta)` 是 Weingarten 变换在 `bm r_u, bm r_v` 下的矩阵,
`delta_alpha^beta` 是 Kroneker 符号.
我们有 `g^(alpha beta) = g^(beta alpha)`, 但一般 `b_alpha^beta !=
b_beta^alpha`.
在一个单项式中, 若一个指标字母 (如 `alpha`) 作为上标和下标各出现一次,
则该式表示对 `alpha = 1, 2` 的求和式;
上下指标多对重复出现就表示该式是多重求和式. 如
`"d"bm r = bm r_1 "d"u^1 + bm r_2 "d"u^2 = bm r_alpha "d"u^alpha`,
`"I" = g_(11)"d"u^1"d"u^1 + 2g_(12)"d"u^1"d"u^2 +
g_(22)"d"u^2"d"u^2 = g_(alpha beta)"d"u^alpha"d"u^beta`,
`"II" = b_(11)"d"u^1"d"u^1 + 2b_(12)"d"u^1"d"u^2 +
b_(22)"d"u^2"d"u^2 = b_(alpha beta)"d"u^alpha"d"u^beta`.
用待定系数法推导曲面 `S` 自然标架 `{bm r";" bm r_1, bm r_2, bm n}` 的运动方程: `{ (del bm r)/(del u^alpha) = bm r_alpha, alpha = 1","2; (del bm r_alpha)/(del u^beta) = Gamma_(alpha beta)^gamma bm r_gamma + b_(alpha beta) bm n, alpha"," beta = 1"," 2; (del bm n)/(del u^alpha) = -b_alpha^beta bm r_beta, alpha = 1","2 :}` 其中 `Gamma_(alpha beta)^gamma` `= 1/2 g^(xi gamma)( (del g_(alpha xi))/(del u^beta) + (del g_(beta xi))/(del u^alpha) - (del g_(alpha beta))/(del u^xi) )`. 可以看到, 自然标架的运动由第一, 第二基本形式的系数完全决定.
设
`(del bm r_alpha)/(del u^beta) = bm r_(alpha beta) = Gamma_(alpha
beta)^gamma bm r_(gamma) + C_(alpha beta) bm n`,
`(del bm n)/(del u^alpha) = bm n_alpha = D_alpha^beta bm r_beta +
D_alpha bm n`.
两边与 `bm n` 作内积得
`C_(alpha beta) = (:bm r_(alpha beta), bm n:) = b_(alpha beta)`.
两边与 `bm n` 作内积得
`D_alpha = (:bm n_alpha, bm n:) = 0`.
两边与 `bm r_gamma` 作内积得
`D_alpha^beta (:bm r_beta, bm r_gamma:) = (:bm n_alpha, bm
r_gamma:) = -b_(alpha gamma)`.
上式两边同乘 `g^(gamma xi)`, 并对 `gamma` 求和, 有
` D_alpha^xi
= D_alpha^beta delta_beta^xi
= D_alpha^beta g_(beta gamma) g^(gamma xi)
= -b_(alpha gamma) g^(gamma xi)
= -b_alpha^xi`.
最后, 分别对 `g_(alpha beta)`, `g_(alpha gamma)`, `g_(beta gamma)`
求偏导有
`(del g_(alpha beta))/(del u^gamma)
= (:bm r_(alpha gamma), bm r_beta:)
+ (:bm r_(beta gamma), bm r_alpha:)`,
`(del g_(alpha gamma))/(del u^beta)
= (:bm r_(alpha beta), bm r_gamma:)
+ (:bm r_(gamma beta), bm r_alpha:)`,
`(del g_(beta gamma))/(del u^alpha)
= (:bm r_(beta alpha), bm r_gamma:)
+ (:bm r_(gamma alpha), bm r_beta:)`.
后两式相加, 减去第一式得
`(:bm r_(alpha beta), bm r_gamma:) = 1/2 (
(del g_(alpha gamma))/(del u^beta)
+ (del g_(beta gamma))/(del u^alpha)
- (del g_(alpha beta))/(del u^gamma) )`.
现在 两边与 `bm r_xi` 作内积得
`Gamma_(alpha beta)^gamma g_(gamma xi) = (:bm r_(alpha beta), bm
r_xi:)`,
于是
` Gamma_(alpha beta)^gamma
= Gamma_(alpha beta)^eta delta_eta^gamma
= Gamma_(alpha beta)^eta g_(eta xi) g^(xi gamma)
= (:bm r_(alpha beta), bm r_xi:) g^(xi gamma)`
`= 1/2 g^(xi gamma)(
(del g_(alpha xi))/(del u^beta)
+ (del g_(beta xi))/(del u^alpha)
- (del g_(alpha beta))/(del u^xi) )`.
`Gamma_(alpha beta)^gamma` 称为 (第一类) Christoffel 符号, 它由曲面第一基本形式的系数及其偏导数完全决定. 又 ` Gamma_(xi alpha beta)` `= g_(gamma xi) Gamma_(alpha beta)^gamma` `= (:bm r_(alpha beta), bm r_xi:)` `= 1/2 ( (del g_(alpha xi))/(del u^beta) + (del g_(beta xi))/(del u^alpha) - (del g_(alpha beta))/(del u^xi) )` 称为曲面的第二类 Christoffel 符号. 可以看出, 两类 Christoffel 符号都关于 `alpha, beta` 对称.
` Gamma_(delta eta gamma) - (del g_(delta eta))/(del u^gamma) = 1/2 ( - (del g_(eta delta))/(del u^gamma) + (del g_(gamma delta))/(del u^eta) - (del g_(eta gamma))/(del u^delta) ) = -Gamma_(eta delta gamma)`, `b_beta^xi Gamma_(xi alpha gamma)` `= b_(beta delta) g^(delta xi) g_(xi eta) Gamma_(alpha gamma)^eta` `= b_(beta eta) Gamma_(alpha gamma)^eta`.
由 式:
` Gamma_(alpha beta)^gamma
= 1/2 g^(xi gamma)(
(del g_(alpha xi))/(del u^beta)
+ (del g_(beta xi))/(del u^alpha)
- (del g_(alpha beta))/(del u^xi) )`
可以求出
`[Gamma_11^1, Gamma_12^1; Gamma_21^1, Gamma_22^1]`
`= 1/(2(EG-F^2)) [G E_u + F E_v - 2F F_u, G E_v - F E_v;
G E_v - F E_v, 2G F_v - G G_u - F G_v]`
`overset ** =
1/2 [(ln E)_u, (ln E)_v; (ln E)_v, -G_u//E]`,
`[Gamma_11^2, Gamma_12^2; Gamma_21^2, Gamma_22^2]`
`= 1/(2(EG-F^2)) [2E F_u - F E_u - E E_v, E G_u - F E_v;
E G_u - F E_v, F G_u + E G_v - 2F F_v]`
`overset ** =
1/2 [-E_v//G, (ln G)_u; (ln G)_u, (ln G)_v]`.
其中当 `(u,v)` 是曲面的正交参数 (即 `F -= 0`) 时, `overset ** =` 成立.
先求矩阵 `(g^(alpha beta))` `= [E, F; F, G]^-1` `= 1/(E G-F^2) [G, -F; -F, E]`. 以 `Gamma_11^1` 为例: `Gamma_11^1` `= 1/2 g^(xi 1) (2(del g_(1 xi))/(del u^1) - (del g_11)/(del u^xi))` `= 1/2 (2 {:g^11:}(del g_11)/(del u^1) - {:g^11:}(del g_11)/(del u^1) + 2 {:g^21:} (del g_12)/(del u^1) - {:g^21:} (del g_11)/(del u^2))` `= 1/(E G-F^2) (1/2 G E_u - F F_u + 1/2 F E_v)`.
单位球面在球极投影参数下
` [E, F; F, G] = 4/(1+u^2+v^2) bm E`,
`[Gamma_11^1, Gamma_12^1; Gamma_21^1, Gamma_22^1]
= 2/(1+u^2+v^2) [-u,-v; -v,u]`,
`[Gamma_11^2, Gamma_12^2; Gamma_21^2, Gamma_22^2]
= 2/(1+u^2+v^2) [v,-u; -u,-v]`.
根据二阶连续可微函数的二阶偏导数可交换次序, 有 `bm r_(alpha beta) = bm r_(beta alpha)`, `bm r_(alpha beta gamma) = bm r_(alpha gamma beta)`, `bm n_(alpha beta) = bm n_(beta alpha)`.
先看 . 联系自然标架的运动方程, 它等价于 `Gamma_(alpha beta)`, `b_(alpha beta)` 关于 `alpha, beta` 对称.
再看 . 因为
`bm r_(alpha beta gamma)`
`= del/(del u^gamma)
(Gamma_(alpha beta)^xi bm r_xi + b_(alpha beta) bm n)`
`= (del Gamma_(alpha beta)^xi)/(del u^gamma) bm r_xi
+ Gamma_(alpha beta)^xi bm r_(xi gamma)
+ (del b_(alpha beta))/(del u^gamma) bm n
+ b_(alpha beta) (del bm n)/(del u^gamma)`
`= (del Gamma_(alpha beta)^xi)/(del u^gamma) bm r_xi
+ Gamma_(alpha beta)^xi
(Gamma_(xi gamma)^eta bm r_eta + b_(xi gamma) bm n)`
`+ (del b_(alpha beta))/(del u^gamma) bm n
+ b_(alpha beta) (-b_gamma^xi bm r_xi)`
`= ( ( del Gamma_(alpha beta)^xi)/(del u^gamma)
+ Gamma_(alpha beta)^eta Gamma_(eta gamma)^xi
- b_(alpha beta) b_gamma^xi ) bm r_xi`
`+ ( Gamma_(alpha beta)^xi b_(xi gamma)
+ (del b_(alpha beta))/(del u^gamma) ) bm n`,
而由
知上式左边关于 `beta, gamma` 对称, 所以上式右边也是如此.
利用 `bm r_1, bm r_2, bm n` 线性无关得到
Gauss 方程:
`b_(alpha beta) b_gamma^xi
- b_(alpha gamma) b_beta^xi`
`= (del Gamma_(alpha beta)^xi)/(del u^gamma)
- (del Gamma_(alpha gamma)^xi)/(del u^beta)`
`+ Gamma_(alpha beta)^eta Gamma_(eta gamma)^xi
- Gamma_(alpha gamma)^eta Gamma_(eta beta)^xi`,
和 Codazzi 方程:
` (del b_(alpha beta))/(del u^gamma)
- (del b_(alpha gamma))/(del u^beta)`
`= - Gamma_(alpha beta)^xi b_(xi gamma)
+ Gamma_(alpha gamma)^xi b_(xi beta)`.
最后看 . 我们有 ` bm n_(beta gamma) = del/(del u^gamma) (b_beta^xi bm r_xi) = (del b_beta^xi)/(del u^gamma) bm r_xi + b_beta^xi(Gamma_(xi gamma)^eta bm r_eta + b_(xi gamma) bm n)` `= ( (del b_beta^xi)/(del u^gamma) + b_beta^eta Gamma_(eta gamma)^xi ) bm r_xi + b_beta^xi b_(xi gamma) bm n`. 同样由上式对 `beta, gamma` 对称有 ` (del b_beta^xi)/(del u^gamma) - (del b_gamma^xi)/(del u^beta) = - b_beta^eta Gamma_(eta gamma)^xi + b_gamma^eta Gamma_(eta beta)^xi`. (由于 `(g^(eta xi))` 对称, `b_beta^xi b_(xi gamma) = b_(beta eta) g^(eta xi) b_(xi gamma)` 是对称矩阵的合同阵, 它关于 `beta, gamma` 对称是显然的). 可以证明上式与 Codazzi 方程是等价的.
计算 `g_(alpha xi) ( (del b_beta^xi)/(del u^gamma) + b_beta^eta Gamma_(eta gamma)^xi )` `= (del b_(alpha beta))/(del u^gamma) - b_beta^xi (del g_(alpha xi))/(del u^gamma) + b_beta^eta Gamma_(alpha eta gamma)` `= (del b_(alpha beta))/(del u^gamma) + b_beta^xi (Gamma_(alpha xi gamma) - (del g_(alpha xi))/(del u^gamma) )` `= (del b_(alpha beta))/(del u^gamma) - b_beta^xi Gamma_(xi alpha gamma)` `= (del b_(alpha beta))/(del u^gamma) - Gamma_(alpha gamma)^xi b_(xi beta)`. 上式两边置换 `beta, gamma` 并相减, 就能联系起 Codazzi 方程的两个等价形式.
由于 `alpha, beta, gamma, xi = 1, 2`, Gauss 方程共有 16 个, Codazzi 方程共 8 个. 不过实质上 Gauss 方程只有 1 个独立方程, Codazzi 方程只有 2 个独立方程. 为说明这一点, 用 `g_(delta xi)` 乘 Gauss 方程左边, 称为 Riemann 记号: `R_(delta alpha beta gamma)` `= g_(delta xi)(b_(alpha beta)b_gamma^xi - b_(alpha gamma)b_beta^xi)` `= b_(alpha beta) b_(gamma delta) - b_(alpha gamma) b_(beta delta)` `= |b_(alpha beta), b_(alpha gamma); b_(beta delta), b_(gamma delta)|`.
置换性质 Riemann 记号满足 ` R_(delta alpha beta gamma)` `= R_(beta gamma delta alpha)` `= -R_(alpha delta beta gamma)` `= -R_(delta alpha gamma beta)`.
Gauss 方程的左边也满足同样的置换性质; 因此 Gauss 方程中只有一个独立方程 `R_1212 = -(b_11 b_22 - b_12^2)`.
Codazzi 方程的 `beta = gamma` 时, 为平凡等式. 于是 Codazzi 方程只有两个独立方程 `{ (del b_11)/(del u^2) - (del b_12)/(del u^1) = b_(1 xi) Gamma_12^xi - b_(2 xi) Gamma_11^xi; (del b_21)/(del u^2) - (del b_22)/(del u^1) = b_(1 xi) Gamma_22^xi - b_(2 xi) Gamma_21^xi; :}`
当 `(u,v)` 为正交参数系, 即 `F -= 0` 时, Gauss 方程化简为 `-1/sqrt(EG) [(((sqrt E)_v)/sqrt G)_v + (((sqrt G)_u)/sqrt E)_u] = (LN-M^2)/(EG)`, Codazzi 方程化简为 `{ (L/sqrt E)_v - (M/sqrt E)_u = N(sqrt E)_v/G + M(sqrt G)_u/sqrt(E G); (N/sqrt G)_u - (M/sqrt G)_v = L(sqrt G)_u/E + M(sqrt E)_v/sqrt(E G); :}` 若 `(u,v)` 为单位正交标架, 两个方程可以继续简化, 见下文.
设 `S: bm r(u^1,u^2)`, `bar S: bm r(u^1,u^2)` 是定义在同一个参数区域 `D` 上的两个曲面, 如果 `S` 和 `bar S` 的第一, 第二基本形式在 `D` 上处处相等, 则两个曲面相差一个 `E^3` 的刚体运动.
设平面区域 `D` 定义了两个二次微分式
`varphi = g_(alpha beta) "d"u^alpha"d"u^beta`,
`psi = b_(alpha beta) "d"u^alpha"d"u^beta`,
且 `(b_(alpha beta))` 是对称阵, `(g_(alpha beta))` 是正定对称阵.
如果 `Gamma_(alpha beta)^gamma`, `b_(alpha beta)`, `b_beta^alpha`
满足 Gauss-Codazzi 方程, 则对 `D` 中任意的一点 `(u_0^1, u_0^2)`, 存在
一个邻域 `U sube D` 及定义在 `U` 上的曲面 `bm r(u^1, u^2)`, 使得
`varphi, psi` 分别为该曲面的第一, 第二基本形式.
自然标架运动方程是一个一阶偏微分线性方程组, Gauss-Codazzi 方程实质上是这个方程组的可积性条件.
引入正交活动标架通常能简化计算.
在曲面 `S` 各点的切平面上选取单位向量 `bm e_1, bm e_2`, 使 `(:bm e_i, bm e_j:) = delta_(ij)`, `i, j = 1, 2`; 再令 `bm e_3 = bm e_1 ^^ bm e_2`, 则 `{bm r";" bm e_1, bm e_2, bm e_3}` 构成曲面的一个正交标架, 或规范标架.
定义参数区域 `D` 上的一阶微分形式 `omega_1, omega_2` 满足
`"d"bm r = omega_1 bm e_1 + omega_2 bm e_2`, 则
`omega_1 = (:"d"bm r, bm e_1:)`,
`omega_2 = (:"d"bm r, bm e_2:)`,
`"I" = (:"d"bm r, "d"bm r:) = omega_1 omega_1 + omega_2 omega_2`.
又定义一阶微分形式 `omega_(ij)` 满足
`"d"bm e_i = sum_(j=1)^3 omega_(ij) bm e_j`,
换言之,
`omega_(i j) = (:"d"bm e_i, bm e_j:)`,
则由这个关于单位向量微分的定理, 有
`omega_(ij) + omega_(ji) = 0`, 故
`"II" = -(:"d"bm r, "d"bm e_3:)`
`= -(:omega_1 bm e_1 + omega_2 bm e_2, omega_31 bm e_1 + omega_32
bm e_2:)`
`= omega_1 omega_13 + omega_2 omega_23`.
称
`"d"[bm r; bm e_1; bm e_2; bm e_3]
= [omega_1, omega_2, 0;
0, omega_12, omega_13;
omega_21, 0, omega_23;
omega_31, omega_32, 0]
[bm e_1; bm e_2; bm e_3]`,
(其中 `omega_(ij) + omega_(ji) = 0`) 为曲面正交标架的运动方程.
由以上定义可以看出, 曲面的 `"I", "II"` 由正交标架运动方程的系数决定. 且可以证明, `"I"` 与正交标架的选取无关; 法向量确定的情况下, `"II"` 也与正交标架选取无关.
由于 `bm r_u, bm r_v` 和 `bm e_1, bm e_2` 都是切平面的基, 可以设 `[bm r_u; bm r_v] = bm A [bm e_1; bm e_2]`, `bm A` 为可逆矩阵. 故 `["d"u, "d"v] bm A = [omega_1, omega_2]`. 我们知道一阶微分形式 `omega_13, omega_23` 是 `"d"u, "d"v` 的线性组合; 由上式, 它也可以表示为 `omega_1, omega_2` 的线性组合. 故存在矩阵 `bm B` 使得 `[omega_13, omega_23] = [omega_1, omega_2] bm B`.
现在来考虑曲面的第一, 第二基本形式. 我们有 ` "I" = [omega_1, omega_2] [omega_1; omega_2] = ["d"u, "d"v] bm(A A)^T ["d"u; "d"v]`, ` "II" = [omega_13, omega_23][omega_1; omega_2] = [omega_1, omega_2] bm B [omega_1; omega_2] = ["d"u, "d"v] bm (A B A)^T ["d"u; "d"v]`. 联系 `"I", "II"` 的定义得 `[E,F; F,G] = bm(A A)^T`, `[L,M; M,N] = bm(A B A)^T`.
上式重新得出了 `[E,F; F,G]` 正定的结论; 由于对称矩阵在合同变换下保持对称, 所以 `bm B` 是对称矩阵.
由
`"d"bm r = omega_1 bm e_1 + omega_2 bm e_2`,
`-"d"bm n = -"d"bm e_3 = omega_13 bm e_1 + omega_23 bm e_2`
`= [omega_1, omega_2] bm B [bm e_1; bm e_2]`
以及 `cc W("d"bm r) = -"d"bm n` 知道, `bm B` 恰为 Weingarten
变换在正交标架下的矩阵.
曲面的主曲率恰为 `bm B` 的两个特征值; Gauss 曲率等于 `|bm B|`;
平均曲率等于 `1/2 "tr"bm B`.
当曲面没有脐点时, 特别取 `bm e_1, bm e_2` 为曲面的主方向, 有
`(:cc W(bm e_1), bm e_1:) = k_1`,
`(:cc W(bm e_2), bm e_2:) = k_2`,
`(:cc W(bm e_1), bm e_2:) = (:cc W(bm e_2), bm e_1:) = 0`,
即 `bm B` 为对角矩阵. 这时 `"II" = k_1 omega_1 omega_1 + k_2 omega_2
omega_2`.
称平面参数区域 `D = {(u, v)}` 上的函数为零阶外微分形式. 零阶微分形式关于 `"d"u, "d"v` 的线性组合, 形如 `f"d"u + g"d"v`, 称为一阶外微分形式. 如 `"d"u`, `"d"v` 都是一阶外微分形式.
现在可以定义微分形式的外微分运算 `"d"`.
` "d"f = (del f)/(del u) "d"u + (del f)/(del v) "d"v`,
` "d"(f"d"u+g"d"v)`
`= "d"f ^^ "d"u + "d"g ^^ "d"v`
`= ((del g)/(del u) - (del f)/(del v)) "d"u ^^ "d"v`,
` "d"(f "d"u ^^ "d"v) = "d"f ^^ "d"u ^^ "d"v = 0`.
记忆: 设 `varphi = f "d"u + g"d"v`, 则 `"d"varphi = |del/(del u), del/(del v); f, g| "d"u ^^ "d"v`. 曲面上区域 `D` 的 Green 公式可以写为 `oint_(del D) varphi = iint_D "d"varphi`.
对正交标架的曲面运动方程第一式
`"d"bm r = omega_1 bm e_1 + omega_2 bm e_2` 两边求外微分,
` bb 0 = "d"(sum_(i=1)^2 omega_i bm e_i)`
`= sum_(i=1)^2 ("d"omega_i bm e_i - omega_i ^^ "d"bm e_i)`
`= sum_(i=1)^2 ("d"omega_i bm e_i - omega_i ^^ sum_(j=1)^3
omega_(ij) bm e_j)`
`= sum_(i=1)^2 "d"omega_i bm e_i - sum_(j=1)^3 sum_(i=1)^2
omega_i ^^ omega_(ij) bm e_j.`
由 `bm e_1, bm e_2, bm e_3` 线性无关知,
`"d"omega_j = sum_(i=1)^2 omega_i ^^ omega_(ij)`, `quad j = 1, 2`;
`0 = sum_(i=1)^2 omega_i ^^ omega_(i3)`.
将 `omega_(i3) = sum_(j=1)^2 h_(ij) omega_j` 代入 , 得
` 0
= sum_(i=1)^2 omega_i ^^ sum_(j=1)^2 h_(ij) omega_j
= sum_(i=1)^2 sum_(j=1)^2 h_(ij) omega_i ^^ omega_j
= (h_12 - h_21) omega_1 ^^ omega_2`.
所以 `h_12 = h_21`,
等价于矩阵 `bm B = (h_(ij))^T` 是对称的.
另一方面, 因为 `omega_11 = omega_22 = 0`,
简化为
`{
"d" omega_1 = omega_2 ^^ omega_21;
"d" omega_2 = omega_1 ^^ omega_12;
:}`.
同样, 对运动方程的第二式
`"d"bm e_i = sum_(j=1)^3 omega_(ij) bm e_j` (`i = 1, 2, 3`) 求外微分,
` bb 0 = "d"(sum_(j=1)^3 omega_(ij) bm e_j)`
`= sum_(j=1)^3 ("d"omega_(ij) bm e_j - omega_(ij) ^^ "d"bm e_j)`
`= sum_(j=1)^3 ("d"omega_(ij) bm e_j
- omega_(ij) ^^ sum_(k=1)^3 omega_(jk) bm e_k)`
`= sum_(j=1)^3 "d"omega_(ij) bm e_j
- sum_(k=1)^3 sum_(j=1)^3 omega_(ij) ^^ omega_(jk) bm e_k`
`= sum_(k=1)^3 ("d"omega_(ik)
- sum_(j=1)^3 omega_(ij) ^^ omega_(jk)) bm e_k`.
因此
`"d"omega_(ik) = sum_(j=1)^3 omega_(ij) ^^ omega_(jk)`,
`quad i, k = 1, 2, 3`.
由于 `omega_(ik)` 的反对称性, `i = k` 时, 上面的等式是平凡的;
将 `i != k` 的情形写出, 即
(Gauss) `"d"omega_12 = omega_13 ^^ omega_32`;
(Codazzi) `{
"d"omega_13 = omega_12 ^^ omega_23;
"d"omega_23 = omega_21 ^^ omega_13;
:}`.
,
,
合称为曲面正交标架的结构方程式,
它们是正交标架运动方程的可积性条件.
下面说明 就是 Gauss 方程, 就是 Codazzi 方程.
设 `(u,v)` 是曲面的正交参数, 取 `bm e_1 = bm r_u/sqrt E`, `bm e_2 = bm
r_v/sqrt G`, 容易验证 `bm e_1, bm e_2` 是切平面的单位正交基.
此时
`bm A = [sqrt E, 0; 0, sqrt G]`,
`bm B = bm A^-1 [L,M; M,N] (bm A^T)^-1
= [L//E, M//sqrt(EG); M//sqrt(EG),N//G]`.
于是 `omega_1 = sqrt E "d"u`, `omega_2 = sqrt G"d"v`.
利用 有
` omega_12 ^^ omega_2 = "d"omega_1 = -(sqrt E)_v "d"u ^^ "d"v
= -(sqrt E)_v/sqrt G "d"u ^^ omega_2`,
` omega_21 ^^ omega_1 = "d"omega_2 = (sqrt G)_u "d"u ^^ "d"v
= -(sqrt G)_u/sqrt E "d"v ^^ omega_1`.
记 `beta = -(sqrt E)_v/sqrt G "d"u`, `gamma = (sqrt G)_u/sqrt E "d"v`,
由上式,
`(omega_12 - beta - gamma) ^^ omega_1 = (omega_12 - gamma) ^^
omega_1 = 0`,
`(omega_12 - beta - gamma) ^^ omega_2 = (omega_12 - beta) ^^
omega_2 = 0`.
但 `omega_1 ^^ omega_2 != 0`, 所以 `omega_12 - beta - gamma = 0`, 即
`omega_12 = beta + gamma = -(sqrt E)_v/sqrt G "d"u + (sqrt
G)_u/sqrt E "d"v`.
再利用 `[omega_13, omega_23] = [omega_1, omega_2] bm B` 得
`omega_13 = L/sqrt E "d"u + M/sqrt E "d"v`,
`omega_23 = M/sqrt G "d"u + N/sqrt G "d"v`.
将上述结果代入 计算,
就得到正交参数下的 Gauss 方程;
代入 ,
就得到正交参数下的 Codazzi 方程.
Gauss 方程 也可以写为 ` "d"omega_12 = -sum_(i=1)^2 h_(1i) omega_i ^^ sum_(j=1)^2 h_(2j) omega_j = -sum_(i=1)^2 sum_(j=1)^2 h_(1i) h_(2j) omega_i ^^ omega_j` `=-(h_11 h_22 - h_12 h_21) omega_1 ^^ omega_2 = -K omega_1 ^^ omega_2`.
设 `{bm r";" bm e_1, bm e_2, bm e_3}` 和 `{bm bar r";" bm bar e_1, bm bar e_2, bm bar e_3}` 是曲面的两组正交标架 (都是右手系), `{omega_1, omega_2, omega_12, omega_13, omega_23}` 和 `{bar omega_1, bar omega_2, bar omega_12, bar omega_13, bar omega_23}` 是相应的诸微分形式. 又设这两个标架间相差一个 `theta` 角的旋转: `[bm bar e_1; bar bm e_2] = bm T [bm e_1; bm e_2]`, `quad bm T = [cos theta, -sin theta; sin theta, cos theta]`, 计算知 `{bar omega_1, bar omega_2}` 与 `{omega_1, omega_2}` 之间, `{bar omega_13, bar omega_23}` 与 `{omega_13, omega_23}` 之间均相差一个相同的旋转: `[bar omega_1; bar omega_2] = bm T [omega_1; omega_2]`, `quad [bar omega_13; bar omega_23] = bm T [omega_13; omega_23]`. 由此可以验证以下的量与正交标架选取无关. 这些不依赖于 (同向) 正交标架选取的量称为曲面的几何量:
`omega_12` 称为联络形式, 我们将在下一章研究曲面的内蕴几何学时进一步讨论. 由 `bar omega_12 = (:"d"bm bar e_1, bm bar e_2:)` 计算知 `bar omega_12 = omega_12 + "d"theta`. 故 `omega_12` 依赖于正交标架的选取, 不是几何量.