1 From Riemann manifolds to Riemann manifolds
3
f
f
M
2
l
M
2
r
Φr
Φ l
˘ R
2 , δ ij
¯
= E
2
˘ R
2 , δ IJ
¯
= E
2
U r ⊂ M
2
r
M
2
l ⊃ U l
Fig. 1.1. Commutative diagram (f, f, Φ l , Φ r ); f : M
2
l → M
2
r ; f = Φ r ◦ f ◦ Φ
−1
l .
Example 1.2 (E
2
A 1 ,A 1 ,A 2
→ S
2
r , isoparametric mapping).
As an example of the mapping f : M
2
l −→ M
2
r and the commutative diagram (f, f, Φ l , Φ r ), think of
an ellipsoid-of-revolution
E
2
A 1 ,A 1 ,A 2
:=
X ∈ R
3 X
2 + Y
2
A 2
1
+
Z
2
A 2
2
= 1 , R
+
A 1 > A 2 ∈ R
+
(1.9)
of semi-major axis A 1 and semi-minor axis A 2 as the left Riemann manifold M
2
l = E
2
A 1 ,A 1 ,A 2
, and
think of a sphere
S
2
r :=
x ∈ R
2
x
2 + y
2 + z
2 = r
2 , r ∈ R
+
(1.10)
of radius r as the right Riemann manifold M
2
r = S
2
r , f being the pointwise mapping of E
2
A 1 ,A 1 ,A 2
to S
2
r one-to-one. f could be illustrated by a transformation of {ellipsoidal longitude Λ, ellipsoidal
latitude Φ} onto {spherical longitude λ, spherical latitude φ} one-to-one. The mapping f = id is
called isoparametric if {Λ = λ, Φ = φ} or {U = u, V = v} in general coordinates of the left Riemann
manifold and the right Riemann manifold, respectively. Accordingly, in an isoparametric mapping,
{ellipsoidal longitude, ellipsoidal latitude} and {spherical longitude, spherical latitude} are identical.
End of Example.
f
f
Λ
λ
Φ
φ
Φ r
Φ l
M
2
l = E
2
A 1 ,A 1 ,A 2
M
2
r = S
2
r
Φ l :=
»
arctan
Y
X
, arctan
A
2
1 A
−2
2
Z
√
X 2 +Y 2
–
Φ r :=
»
arctan
y
x
, arctan
z
√
x 2 +y 2
–
Fig. 1.2. Bijective mapping of an ellipsoid-of-revolution E
2
A 1 ,A 1 ,A 2 to a sphere S
2
r ; f : E
2
A 1 ,A 1 ,A 2 → S
2
r ;
Φ l := [Λ, Φ], Φ r := [λ, φ]; isoparametric mapping f = id, namely {Λ, Φ} = {λ, φ}.