2 From Riemann manifolds to Euclidean manifolds
Mapping from a left two-dimensional Riemann manifold to a right two-dimensional Euclidean manifold,
Cauchy–Green and Euler–Lagrange deformation tensors, equivalence theorem for equiareal mappings,
conformeomorphism and areomorphism, Korn–Lichtenstein equations and Cauchy–Riemann equations,
Mollweide projection, canonical criteria for (conformal, equiareal, isometric, equidistant) mappings, polar
decomposition and simultaneous diagonalization for more than two matrices.
Let there be given the left two-dimensional Riemann manifold {M
2
l , G MN } as well as the right twodimensional Euclidean manifold {M
2
r , g µν } = {R
2 , δ µν } = E
2 . In many applications, the choice of
{R
2 , δ µν } is the “plane manifold”, for instance, (i) the equatorial plane of the sphere or the ellipsoid,
(ii) the meta-equatorial, also called oblique equatorial plane of the sphere or the ellipsoid, (iii) the
plane generated by developing the cylinder, the cone, a ruled surface (namely surfaces which are
“Gauss flat”), (iv) the tangent space T U 0 M
2
l of the left two-dimensional Riemann manifold fixed to
the point U 0 := {U
1
0 , U
2
0 } being covered by Cartesian coordinates. (Refer to all previous examples.)
We shall not repeat the various deformation measures of type multiplicative and additive for the
special case of the right two-dimensional Euclidean manifold {R
2 , δ µν }. Instead, we present to you
(i) the left and right eigenspace analysis and synthesis of the Cauchy–Green deformation tensor,
special case {M
2
r , g µν } = {R
2 , δ µν }, (ii) the left and right eigenspace analysis and synthesis of the
Euler–Lagrange deformation tensor, special case {M
2
r , g µν } = {R
2 , δ µν }, (iii) conformeomorphism,
conformal mapping, special case {M
2
r , g µν } = {R
2 , δ µν }; Korn–Lichtenstein equations, special case
Cauchy–Riemann equations (d’Alembert–Euler equations).
2-1 Eigenspace analysis, Cauchy–Green deformation tensor
Left and right eigenspace analysis and synthesis of the Cauchy–Green deformation tensor, special case
{M
2
r , g µν } = {R
2 , δ µν }.
First, let us confront you with Lemma 2.1, where we present detailed results of the left and right
eigenspace analysis and synthesis of the Cauchy–Green deformation tensor for the special case of a
right Euclidean manifold. Second, we focus on an interpretation of the results and additionally discuss
a short example.
Lemma 2.1 (Left and right eigenspace analysis and synthesis of the Cauchy–Green deformation tensor,
special case {M
2
r , g µν } = {R
2 , δ µν }).
(i) Synthesis.
For the matrix pair of positive-definite and symmetric matrices {C l , G l } or {C r , G r }, a simultaneous
diagonalization is (the right Frobenius matrix F r is an orthonormal matrix)
C l = J
T
l J l , F
T
l C l F l = diag
Λ
2
1 , Λ
2
2
, F
T
l G l F l = I versus F
T
r C r F r = diag
λ
2
1 , λ
2
2
, F
T
r F r = I . (2.1)
(ii) Analysis.
Left eigenvalues or left principal stretches:
C l − Λ
2
i G l
= 0 ,
Λ
2
1,2 = Λ
2
± =
1
2
tr
C l G
−1
l
±
tr
C l G
−1
l
2 − 4det
C l G
−1
l
.
(2.2)
Mapping from a left two-dimensional Riemann manifold to a right two-dimensional Euclidean manifold,
Cauchy–Green and Euler–Lagrange deformation tensors, equivalence theorem for equiareal mappings,
conformeomorphism and areomorphism, Korn–Lichtenstein equations and Cauchy–Riemann equations,
Mollweide projection, canonical criteria for (conformal, equiareal, isometric, equidistant) mappings, polar
decomposition and simultaneous diagonalization for more than two matrices.
Let there be given the left two-dimensional Riemann manifold {M
2
l , G MN } as well as the right twodimensional Euclidean manifold {M
2
r , g µν } = {R
2 , δ µν } = E
2 . In many applications, the choice of
{R
2 , δ µν } is the “plane manifold”, for instance, (i) the equatorial plane of the sphere or the ellipsoid,
(ii) the meta-equatorial, also called oblique equatorial plane of the sphere or the ellipsoid, (iii) the
plane generated by developing the cylinder, the cone, a ruled surface (namely surfaces which are
“Gauss flat”), (iv) the tangent space T U 0 M
2
l of the left two-dimensional Riemann manifold fixed to
the point U 0 := {U
1
0 , U
2
0 } being covered by Cartesian coordinates. (Refer to all previous examples.)
We shall not repeat the various deformation measures of type multiplicative and additive for the
special case of the right two-dimensional Euclidean manifold {R
2 , δ µν }. Instead, we present to you
(i) the left and right eigenspace analysis and synthesis of the Cauchy–Green deformation tensor,
special case {M
2
r , g µν } = {R
2 , δ µν }, (ii) the left and right eigenspace analysis and synthesis of the
Euler–Lagrange deformation tensor, special case {M
2
r , g µν } = {R
2 , δ µν }, (iii) conformeomorphism,
conformal mapping, special case {M
2
r , g µν } = {R
2 , δ µν }; Korn–Lichtenstein equations, special case
Cauchy–Riemann equations (d’Alembert–Euler equations).
2-1 Eigenspace analysis, Cauchy–Green deformation tensor
Left and right eigenspace analysis and synthesis of the Cauchy–Green deformation tensor, special case
{M
2
r , g µν } = {R
2 , δ µν }.
First, let us confront you with Lemma 2.1, where we present detailed results of the left and right
eigenspace analysis and synthesis of the Cauchy–Green deformation tensor for the special case of a
right Euclidean manifold. Second, we focus on an interpretation of the results and additionally discuss
a short example.
Lemma 2.1 (Left and right eigenspace analysis and synthesis of the Cauchy–Green deformation tensor,
special case {M
2
r , g µν } = {R
2 , δ µν }).
(i) Synthesis.
For the matrix pair of positive-definite and symmetric matrices {C l , G l } or {C r , G r }, a simultaneous
diagonalization is (the right Frobenius matrix F r is an orthonormal matrix)
C l = J
T
l J l , F
T
l C l F l = diag
Λ
2
1 , Λ
2
2
, F
T
l G l F l = I versus F
T
r C r F r = diag
λ
2
1 , λ
2
2
, F
T
r F r = I . (2.1)
(ii) Analysis.
Left eigenvalues or left principal stretches:
C l − Λ
2
i G l
= 0 ,
Λ
2
1,2 = Λ
2
± =
1
2
tr
C l G
−1
l
±
tr
C l G
−1
l
2 − 4det
C l G
−1
l
.
(2.2)