ds2 = dr2 + (sin r)2dθ2. (r, θ) = (x1, x2).
gij = |
| . |
Γ121Γ111
+ Γ122Γ211 = 0
Γ121Γ112
+ Γ122Γ212 = −cos(r)2
(+=> R1221; −=> R1212)
Γ121Γ121
+ Γ122Γ221 = −cos(r)2
(+=> R1122; −=> R1122)
Γ121Γ122
+ Γ122Γ222 = 0
Γ211Γ111
+ Γ212Γ211 = 0
Γ211Γ112
+ Γ212Γ212 = cot(r)2
(+=> R2211; −=> R2211)
Γ211Γ121
+ Γ212Γ221 = cot(r)2
(+=> R2112; −=> R2121)
Γ211Γ122
+ Γ212Γ222 = 0
Γ221Γ111
+ Γ222Γ211 = 0
Γ221Γ112
+ Γ222Γ212 = 0
Γ221Γ121
+ Γ222Γ221 = 0
Γ221Γ122
+ Γ222Γ222 = −cos(r)2
(+=> R2222; −=> R2222)
In summary:
R1212
= sin(r)2 − cos(r)2 + cos(r)2 = sin(r)2
R1221 = cos(r)2 − sin(r)2
−cos(r)2 = − sin(r)2
R2112 = −1 − cot(r)2 + cot(r)2
= −1
R2121 = 1 + cot(r)2 − cot(r)2
= 1
A plausible symmetry.
Also R1212 = R2121 = sin(r)2
= − R2112 = − R1221.
Rij = Rkikj = |
| . |