66
1 From Riemann manifolds to Riemann manifolds
Box 1.33 (Right Korn–Lichtenstein equations, UPS of S
2
r , harmonicity, orientation).
Right Korn–Lichtenstein equations:
(1st) p λ = +
r g 11
g 22
q φ , p φ = −
r g 22
g 11
q λ (2nd) ,
(1st) q λ = −
r g 11
g 22
p φ , q φ = +
r g 22
g 11
p λ (2nd) ;
(1.236)
r g 11
g 22
= cos φ ,
r g 22
g 11
=
1
cos φ
;
(1.237)
(UPS right) q φ = g
(φ) sin λ , p λ = −g(φ) sin λ ,
(UPS right) q λ = g(φ) cos λ , p φ = +g
(φ) cos λ ,
g
(φ) = −
g(φ)
cos φ
⇒
(KL 2nd) g
(φ) sin λ = −
g(φ)
cos φ
sin λ q. e. d.
(KL 1st) g(φ) cos λ = − cos φ g
(φ) cos λ q. e. d.
(1.238)
Right integrability conditions:
∆ λ,φ p =
„r
g 11
g 22
p φ
«
φ
+
„r
g 22
g 11
p λ
«
λ
= 0 ,
∆ λ,φ q =
„r
g 11
g 22
q φ
«
φ
+
„r
g 22
g 11
q λ
«
λ
= 0 .
(1.239)
(1st)
r g 11
g 22
p φ = cos φ g
(φ) cos λ = −g(φ) cos λ ,
r g 22
g 11
p λ =
1
cos φ
[−g(φ) sin λ] = −
g(φ)
cos φ
sin λ
⇒
∆ λ,φ p = −g
(φ) cos λ −
g(φ)
cos φ
cos λ , g
(φ) = −
g(φ)
cos φ
⇒ ∆ λ,φ p = 0 q. e. d.
(1.240)
(2nd)
r g 11
g 22
q φ = cos φ g
(φ) sin λ = −g(φ) sin λ ,
r g 22
g 11
q λ =
1
cos φ
g(φ) cos λ
⇒
∆ λ,φ q = −g
(φ) sin λ −
g(φ)
cos φ
sin λ , g
(φ) = −
g(φ)
cos φ
⇒ ∆ λ,φ q = 0 q. e. d.
(1.241)
Right orientation:
˛
˛
˛
˛
˛
p λ p φ
q λ q φ
˛
˛
˛
˛
˛
= p λ q φ − p φ q λ = −g(φ)g
(φ) =
g
2 (φ)
cos φ
> 0
(1.242)
due to
−π/2 < φ < +π/2 ⇒ cos φ > 0 q.e.d.
(1.243)