58
1 From Riemann manifolds to Riemann manifolds
Box 1.26 (Right Korn–Lichtenstein equations, UMP 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) ;
(1.204)
r g 11
g 22
= cos φ ,
r g 22
g 11
=
1
cos φ
;
(1.205)
(UMP right) q φ =
r
cos φ
, p λ = r (KL 2nd) ,
(UMP right) q λ = 0 , p φ = 0 (KL 1st) .
(1.206)
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 ;
(1.207)
(1st)
r g 11
g 22
p φ = 0 ,
r g 22
g 11
p λ =
r
cos φ
⇒
∆ λ,φ p =
„r
g 11
g 22
p φ
«
φ
+
„r
g 22
g 11
p λ
«
λ
=
„
r
cos φ
«
λ
= 0
(1.208)
q. e. d.
(2nd)
r g 11
g 22
q φ = r ,
r g 22
g 11
q λ = 0
⇒
∆ λ,φ q =
„r
g 11
g 22
q φ
«
φ
+
„r
g 22
g 11
q λ
«
λ
= 0
(1.209)
q. e. d.
Right orientation:
˛
˛
˛
˛
˛
p λ p φ
q λ q φ
˛
˛
˛
˛
˛
= p λ q φ − p φ q λ =
r
2
cos φ
> 0
(1.210)
due to
−π/2 < Φ < +π/2 ⇒ cos Φ > 0
(1.211)
q. e. d.