110 2 From Riemann manifolds to Euclidean manifolds
Box 2.5 (Left Cauchy–Green matrix, generalized Mollweide projection of the ellipsoid-of-revolution).
Left Jacobi matrix:
J l :=
"
D Λ x D Φ x
D Λ y D Φ y
#
,
D Λ x = a cos t , D Φ x = D t xD Φ t = −aΛ sin t t
,
D Λ y = 0 , D Φ y = D t yD Φ t = +b cos t t
.
(2.59)
Left Cauchy–Green matrix:
C l := J
∗
l G r J l , G r = I 2 ⇒ C l = J
∗
l J l ,
C l =
"
a
2 cos
2 t
−aΛ
2 cos t sin t t
−aΛ
2 cos t sin t t
(a
2 Λ
2 sin
2 t + b
2 cos
2 t)(t
)
2
#
.
(2.60)
Left matrix of the metric:
G l =
"
N
2 (Φ)
0
0
M
2 (Φ)
#
(N (Φ) and M (Φ): see Example 1.3).
(2.61)
det
ˆ
C l G
−1
l
˜
= 1:
det
ˆ
C l G
−1
l
˜
=
a
2 cos
2 t
G 11
a
2 Λ
2 sin
2 t + b
2 cos
2 t
G 22
(t
)
2 −
a
4 Λ
2 cos
2 t sin
2 t
G 11 G 22
(t
)
2 =
=
cos
4 t
G 11 G 22
a
2 b
2 (t
)
2 .
(2.62)
(t
):
2(1 + cos 2t)dt =
π
ln
1 + E
1 − E
+
2E
1 − E 2
×
×
"
1 − E sin Φ
1 + E sin Φ
E cos Φ
1 − E sin Φ
+
E cos Φ(1 + E sin Φ)
(1 − E sin Φ) 2
!
+
2E cos Φ
1 − E 2 sin
2 Φ
+
4E
3 sin
2 Φ cos Φ
(1 − E 2 sin
2 Φ) 2
#
dΦ ,
1 + cos 2t = 2 cos
2 t ,
cos
2 t (t
) =
E cos Φ
(1 − E 2 sin
2 Φ) 2
π
ln
1+E
1−E
+
2E
1−E 2
, cos
4 t (t
)
2 =
E
2 cos
2 Φ
(1 − E 2 sin
2 Φ) 4
π
2
“
ln
1+E
1−E
+
2E
1−E 2
” 2 ,
1
G 11 G 22
=
(1 − E
2 sin
2 Φ)
4
A
2
1 cos 2 ΦA
2
1 (1 − E) 2 , a
2 b
2 =
A
4
1 (1 − E
2 )
2
π 2 E 2
„
ln
1 + E
1 − E
+
2E
1 − E 2
« 2
.
(2.63)
(6th) Determinantal identity:
det
ˆ
C l G
−1
l
˜
= 1 .
(2.64)