64
1 From Riemann manifolds to Riemann manifolds
Solution (the first problem).
Start from the conformal map defined in Box 1.31. The three conditions to be fulfilled are given in
Box 1.32 for “left UPS” and in Box 1.33 for “right UPS”. First, we specify the left and the right Korn–
Lichtenstein equations, namely
G 11 /G 22 ,
G 22 /G 11 and
g 11 /g 22 ,
g 22 /g 11 , respectively.
Indeed, by transforming {Q Λ , Q Φ , P Λ , P Φ } as well as {q λ , q φ , p λ , p φ } left and right, KL 1st and KL 2nd
are satified. Second, we analyze the left and the right Laplace–Beltrami equations as the integrability
conditions of the left and the right Korn–Lichtenstein equations, namely by {G 11 , G 22 , P Λ , P Φ , Q Λ , Q Φ }
as well as {g 11 , g 22 , p λ , p φ , q λ , q φ } of E
2
A 1 ,A 1 ,A 2
and S
2
r , respectively. Finally, we succeed to prove that
{P, Q} are left harmonic coordinates and {p, q} are right harmonic coordinates. Third, we prove left
and right orientation by computing the left and right Jacobians which are notably positive.
End of Solution (the first problem).
Solution (the second problem).
Again, we have to refer to standard textbooks of Differential Geometry, where you will find the
representation of the Gaussian curvature of a surface in terms of conformal coordinates (isometric,
isothermal). Let the left and the right matrix of the metric be equipped with a conformally flat
structure {G l = λ
2
l I 2 , G r = λ
2
r I 2 }, which is generated by a left and a right conformal coordinate
representation. Then the left Gaussian curvature and the right Gaussian curvature are provided by
k l = −(1/2λ
2
l )∆ l ln λ
2
l = −(1/λ
2
l )∆ l ln λ l and k r = −(1/2λ
2
r )∆ r ln λ
2
r = −(1/λ
2
r )∆ r ln λ r as well as
∆ l := D P P + D QQ = D
2
P + D
2
Q and D
2
p + D
2
q = D pp + D qq := ∆ r , where ∆ l and ∆ r represent the
left Laplace–Beltrami operator and the right Laplace–Beltrami operator. Let us apply this result in
solving the second problem again.
• By means of Box 1.34, we have outlined how to generate a conformally flat metric of an ellipsoid-ofrevolution and of a sphere. First, we depart from the arc lengths “left dS
2 ” given in “left coordinates”
{Λ, Φ} as well as “right ds
2 ” given in “right coordinates” {λ, φ}. Second, we compute the left and
right Cauchy–Green matrices from “left dP
2 +dQ
2 ” and “right dp
2 +dq
2 ”, the arc lengths squared of
the projective plane covered by left conformal coordinates {P, Q} and by right conformal coordinates
{p, q}, respectively. In particular, we arrive at the two equations dP
2 +dQ
2 = f
2 (Φ)dΛ
2 +f
2 (Φ)dΦ
2
and dp
2 +dq
2 = g
2 (φ)dλ
2 +g
2 (φ)dφ
2 , and the corresponding elements of the left Cauchy–Green matrix C l and of the right Cauchy–Green matrix C r . Third, we determine the left eigenvalues { l Λ
2
1 , l Λ
2
2 }
and the right eigenvalues { r Λ
2
1 , r Λ
2
2 } in solving the left characteristic equation
C l − Λ
2
l G l
= 0
and the right characteristic equation
C r − Λ
2
r G r
= 0. In particular, we prove the identities “left
l Λ
2
1 = l Λ
2
2 = Λ
2
l ” and “right r Λ
2
1 = r Λ
2
2 = Λ
2
r ”, characteristic for a conformal mapping. Fourth, due
to the duality relations Λ
2
l λ
2
l = 1 Λ
2
r λ
2
r = 1, we are able to compute λ
2
l and λ
2
r , respectively, and the
conformally flat metric of type “left dS
2 = λ
2
l (dP
2 +dQ
2 )” and of type “right ds
2 = λ
2
r (dp
2 +dq
2 )”.
• Fifth, by means of Box 1.35, we aim at representing the factors of conformality, λ
2
l (φ) and λ
2
r (φ),
in terms of left conformal coordinates {P, Q} and right conformal coordinates {p, q}. We begin
with transforming the right factor of conformality, λ
2
r (φ) → λ
2
r (p, q), since it is available in closed
form. In contrast, the transformation of the left factor of conformality, λ
2
l (Φ) → λ
2
l (P, Q), is only
symbollically written since f
−1 (
P 2 + Q 2 ) is not available in closed form. Note the beautiful
transformations {sin φ, cos φ} and {sin λ, cos λ} as functions of the “UPS coordinates” p and q.
• Sixth, Box 1.36 outlines that λ
2
r and λ
2
l , respectively, fulfill the conformal representation of the right
and the left Gaussian curvature, here written in two versions as a special Helmholtz differential
equation. The simple representation is performed first, followed by the left representation. For a given
Gaussian curvature k r = 1/r
2 = constant of the sphere S
2
r and k l = (1 − E
2 sin
2 Φ)
2 /[A
2
1 (1 − E
2 )]
of the ellipsoid-of-revolution E
2
A 1 ,A 1 ,A 2
being transformed into {p, q} and {P, Q} left and right
conformal coordinates, we succeed to prove ∆ ln λ
2 + 2kλ
2 = 0 of type “right” and “left”.
End of Solution (the second problem).
1 From Riemann manifolds to Riemann manifolds
Solution (the first problem).
Start from the conformal map defined in Box 1.31. The three conditions to be fulfilled are given in
Box 1.32 for “left UPS” and in Box 1.33 for “right UPS”. First, we specify the left and the right Korn–
Lichtenstein equations, namely
G 11 /G 22 ,
G 22 /G 11 and
g 11 /g 22 ,
g 22 /g 11 , respectively.
Indeed, by transforming {Q Λ , Q Φ , P Λ , P Φ } as well as {q λ , q φ , p λ , p φ } left and right, KL 1st and KL 2nd
are satified. Second, we analyze the left and the right Laplace–Beltrami equations as the integrability
conditions of the left and the right Korn–Lichtenstein equations, namely by {G 11 , G 22 , P Λ , P Φ , Q Λ , Q Φ }
as well as {g 11 , g 22 , p λ , p φ , q λ , q φ } of E
2
A 1 ,A 1 ,A 2
and S
2
r , respectively. Finally, we succeed to prove that
{P, Q} are left harmonic coordinates and {p, q} are right harmonic coordinates. Third, we prove left
and right orientation by computing the left and right Jacobians which are notably positive.
End of Solution (the first problem).
Solution (the second problem).
Again, we have to refer to standard textbooks of Differential Geometry, where you will find the
representation of the Gaussian curvature of a surface in terms of conformal coordinates (isometric,
isothermal). Let the left and the right matrix of the metric be equipped with a conformally flat
structure {G l = λ
2
l I 2 , G r = λ
2
r I 2 }, which is generated by a left and a right conformal coordinate
representation. Then the left Gaussian curvature and the right Gaussian curvature are provided by
k l = −(1/2λ
2
l )∆ l ln λ
2
l = −(1/λ
2
l )∆ l ln λ l and k r = −(1/2λ
2
r )∆ r ln λ
2
r = −(1/λ
2
r )∆ r ln λ r as well as
∆ l := D P P + D QQ = D
2
P + D
2
Q and D
2
p + D
2
q = D pp + D qq := ∆ r , where ∆ l and ∆ r represent the
left Laplace–Beltrami operator and the right Laplace–Beltrami operator. Let us apply this result in
solving the second problem again.
• By means of Box 1.34, we have outlined how to generate a conformally flat metric of an ellipsoid-ofrevolution and of a sphere. First, we depart from the arc lengths “left dS
2 ” given in “left coordinates”
{Λ, Φ} as well as “right ds
2 ” given in “right coordinates” {λ, φ}. Second, we compute the left and
right Cauchy–Green matrices from “left dP
2 +dQ
2 ” and “right dp
2 +dq
2 ”, the arc lengths squared of
the projective plane covered by left conformal coordinates {P, Q} and by right conformal coordinates
{p, q}, respectively. In particular, we arrive at the two equations dP
2 +dQ
2 = f
2 (Φ)dΛ
2 +f
2 (Φ)dΦ
2
and dp
2 +dq
2 = g
2 (φ)dλ
2 +g
2 (φ)dφ
2 , and the corresponding elements of the left Cauchy–Green matrix C l and of the right Cauchy–Green matrix C r . Third, we determine the left eigenvalues { l Λ
2
1 , l Λ
2
2 }
and the right eigenvalues { r Λ
2
1 , r Λ
2
2 } in solving the left characteristic equation
C l − Λ
2
l G l
= 0
and the right characteristic equation
C r − Λ
2
r G r
= 0. In particular, we prove the identities “left
l Λ
2
1 = l Λ
2
2 = Λ
2
l ” and “right r Λ
2
1 = r Λ
2
2 = Λ
2
r ”, characteristic for a conformal mapping. Fourth, due
to the duality relations Λ
2
l λ
2
l = 1 Λ
2
r λ
2
r = 1, we are able to compute λ
2
l and λ
2
r , respectively, and the
conformally flat metric of type “left dS
2 = λ
2
l (dP
2 +dQ
2 )” and of type “right ds
2 = λ
2
r (dp
2 +dq
2 )”.
• Fifth, by means of Box 1.35, we aim at representing the factors of conformality, λ
2
l (φ) and λ
2
r (φ),
in terms of left conformal coordinates {P, Q} and right conformal coordinates {p, q}. We begin
with transforming the right factor of conformality, λ
2
r (φ) → λ
2
r (p, q), since it is available in closed
form. In contrast, the transformation of the left factor of conformality, λ
2
l (Φ) → λ
2
l (P, Q), is only
symbollically written since f
−1 (
P 2 + Q 2 ) is not available in closed form. Note the beautiful
transformations {sin φ, cos φ} and {sin λ, cos λ} as functions of the “UPS coordinates” p and q.
• Sixth, Box 1.36 outlines that λ
2
r and λ
2
l , respectively, fulfill the conformal representation of the right
and the left Gaussian curvature, here written in two versions as a special Helmholtz differential
equation. The simple representation is performed first, followed by the left representation. For a given
Gaussian curvature k r = 1/r
2 = constant of the sphere S
2
r and k l = (1 − E
2 sin
2 Φ)
2 /[A
2
1 (1 − E
2 )]
of the ellipsoid-of-revolution E
2
A 1 ,A 1 ,A 2
being transformed into {p, q} and {P, Q} left and right
conformal coordinates, we succeed to prove ∆ ln λ
2 + 2kλ
2 = 0 of type “right” and “left”.
End of Solution (the second problem).