28
1 From Riemann manifolds to Riemann manifolds
Box 1.16 (Orthogonal projection S
2
R + onto P
2
O , polar coordinates, the second problem).
G r =
»
r
2
0
0 1
–
,
C r according to Box 1.14.
(1.105)
Right general eigenvalue problem:
˛
˛ C r − λ
2 G r
˛
˛ = 0 ⇔ λ
2
1,2 = λ
2
+,− =
1
2
„
tr
ˆ
C r G
−1
r
˜ ±
q `
tr
ˆ
C r G
−1
r
˜´ 2 − 4det
ˆ
C r G
−1
r
˜
«
,
G
−1
r
=
" 1
r 2 0
0 1
#
, C r G
−1
r
=
2
4
1
0
0
R
2
R 2 − r 2
3
5 ,
tr
ˆ
C r G
−1
r
˜
=
2R
2 − r
2
R 2 − r 2 , det
ˆ
C r G
−1
r
˜
=
R
2
R 2 − r 2 ,
q `
tr
ˆ
C r G
−1
r
˜´ 2 − 4det
ˆ
C r G
−1
r
˜
=
r
2
R 2 − r 2 ,
λ
2
1 = λ
2
+ =
R
2
R 2 − r 2 , λ 1 = λ + = +
R
√
R 2 − r 2
, λ
2
2 = λ
2
− = 1 , λ 2 = λ − = +1 .
(1.106)
Alternative solution, right general eigenvalue problem:
C r = diag
„
r
2 ,
R
2
R 2 − r 2
«
, G r = diag
`
r
2 , 1
´
,
˛
˛ C r − λ
2 G r
˛
˛ = 0
⇔
˛
˛
˛
˛
˛
r
2 (1 − λ
2 )
0
0
R
2
R 2 −r 2 − λ
2
˛
˛
˛
˛
˛
=
˛
˛
˛
˛
C 11 − λ
2 g 11
0
0
C 22 − λ
2 g 22
˛
˛
˛
˛ = 0
⇔
C 11 − λ
2 g 11 = r
2 (1 − λ
2 ) = 0 , C 22 − λ
2 g 22 =
R
2
R 2 − r 2 − λ
2 = 0
⇒
λ
2
1 = λ
2
+ =
R
2
R 2 − r 2 , λ 1 = λ + = +
R
√
R 2 − r 2
, λ
2
2 = λ
2
− = 1 , λ 2 = λ − = +1 .
(1.107)
Right eigencolumns:
"
f 11
f 21
#
=
1
√
g 11
"
1
0
#
=
1
r
"
1
0
#
,
"
f 12
f 22
#
=
1
√
g 22
"
0
1
#
=
"
0
1
#
.
(1.108)
Right eigenvectors:
g 1 := g α = D α x = −e 1 r sin α + e 2 r cos α , g 2 := g r = D r x = +e 1 cos α + e 2 sin α ;
(1.109)
1st eigenvector: f α := g α f 11 + g r f 21 , f α (r) = g α
1
r
= −e 1 sin α + e 2 cos α
(tangent vector of image of parallel circles) ;
2nd eigenvector: f r := g α f 12 + g r f 22 , f r (r) = g r
(tangent vector of image of meridians) .
(1.110)