1 From Riemann manifolds to Riemann manifolds
“It is vain to do with more what can be done with fewer.”
(Entities should not be multiplied without necessity.)
William of Ockham (1285-1349)
Mappings from a left two-dimensional Riemann manifold to a right two-dimensional Riemann manifold,
simultaneous diagonalization of two matrices, mappings (isoparametric, conformal, equiareal, isometric,
equidistant), measures of deformation (Cauchy–Green deformation tensor, Euler–Lagrange deformation
tensor, stretch, angular shear, areal distortion), decompositions (polar, singular value), equivalence theorems of conformal and equiareal mappings (conformeomorphism, areomorphism), Korn–Lichtenstein
equations, optimal map projections.
There is no chance to map a curved surface (left Riemann manifold), which differs from a developable
surface to a plane or to another curved surface (right Riemann manifold), without distortion or
deformation. Such distortion or deformation measures are reviewed here as they have been developed
in differential geometry, continuum mechanics, and mathematical cartography. The classification of
various mappings from one Riemann manifold (called left) onto another Riemann manifold (called
right) is conventionally based upon a comparison of the metric.
Example 1.1 (Classification).
The terms equidistant, equiareal, conformal, geodesic, loxodromic, concircular, and harmonic represent
examples for such classifications.
End of Example.
In terms of the geometry of surfaces, this is taking reference to its first fundamental form, namely
the Gaussian differential invariant. In particular, in order to derive certain invariant measures of such
mappings outlined in the frontline examples and called deformation measures, a “canonical formalism”
is applied. The simultaneous diagonalization of two symmetric matrices here is of focal interest. Such
a diagonalization rests on the following Theorem 1.1.
Theorem 1.1 (Simultaneous diagonalization of two symmetric matrices).
If A ∈ R
n×n is a symmetric matrix and B ∈ R
n×n is a symmetric positive-definite matrix such that
the product AB
−1 exists, then there exists a non-singular matrix X such that both following matrices
are diagonal matrices, where I n is the n-dimensional unit matrix:
X
T AX = diag(λ 1 , . . . , λ n ) , X
T BX = I n = diag(1, . . . , 1) .
(1.1)
End of Theorem.
According to our understanding, the theorem had been intuitively applied by C. F. Gauss when he
developed his theory of curvature of parameterized surfaces (two-dimensional Riemann manifold).
Here, the second fundamental form (Hesse matrix of second derivatives, symmetric matrix H) had been
analyzed with respect to the first fundamental form (a product of Jacobi matrices of first derivatives,
a symmetric and positive-definite matrix G). Equivalent to the simultaneous diagonalization of a
symmetric matrix H and a symmetric and positive-definite matrix G is the general eigenvalue problem
|H − λG| = 0 ,
(1.2)
which corresponds to the special eigenvalue problem
HG
−1
− λI n
= 0 ,
(1.3)
where HG
−1 defines the Gaussian curvature matrix
−K = HG
−1 .
(1.4)
“It is vain to do with more what can be done with fewer.”
(Entities should not be multiplied without necessity.)
William of Ockham (1285-1349)
Mappings from a left two-dimensional Riemann manifold to a right two-dimensional Riemann manifold,
simultaneous diagonalization of two matrices, mappings (isoparametric, conformal, equiareal, isometric,
equidistant), measures of deformation (Cauchy–Green deformation tensor, Euler–Lagrange deformation
tensor, stretch, angular shear, areal distortion), decompositions (polar, singular value), equivalence theorems of conformal and equiareal mappings (conformeomorphism, areomorphism), Korn–Lichtenstein
equations, optimal map projections.
There is no chance to map a curved surface (left Riemann manifold), which differs from a developable
surface to a plane or to another curved surface (right Riemann manifold), without distortion or
deformation. Such distortion or deformation measures are reviewed here as they have been developed
in differential geometry, continuum mechanics, and mathematical cartography. The classification of
various mappings from one Riemann manifold (called left) onto another Riemann manifold (called
right) is conventionally based upon a comparison of the metric.
Example 1.1 (Classification).
The terms equidistant, equiareal, conformal, geodesic, loxodromic, concircular, and harmonic represent
examples for such classifications.
End of Example.
In terms of the geometry of surfaces, this is taking reference to its first fundamental form, namely
the Gaussian differential invariant. In particular, in order to derive certain invariant measures of such
mappings outlined in the frontline examples and called deformation measures, a “canonical formalism”
is applied. The simultaneous diagonalization of two symmetric matrices here is of focal interest. Such
a diagonalization rests on the following Theorem 1.1.
Theorem 1.1 (Simultaneous diagonalization of two symmetric matrices).
If A ∈ R
n×n is a symmetric matrix and B ∈ R
n×n is a symmetric positive-definite matrix such that
the product AB
−1 exists, then there exists a non-singular matrix X such that both following matrices
are diagonal matrices, where I n is the n-dimensional unit matrix:
X
T AX = diag(λ 1 , . . . , λ n ) , X
T BX = I n = diag(1, . . . , 1) .
(1.1)
End of Theorem.
According to our understanding, the theorem had been intuitively applied by C. F. Gauss when he
developed his theory of curvature of parameterized surfaces (two-dimensional Riemann manifold).
Here, the second fundamental form (Hesse matrix of second derivatives, symmetric matrix H) had been
analyzed with respect to the first fundamental form (a product of Jacobi matrices of first derivatives,
a symmetric and positive-definite matrix G). Equivalent to the simultaneous diagonalization of a
symmetric matrix H and a symmetric and positive-definite matrix G is the general eigenvalue problem
|H − λG| = 0 ,
(1.2)
which corresponds to the special eigenvalue problem
HG
−1
− λI n
= 0 ,
(1.3)
where HG
−1 defines the Gaussian curvature matrix
−K = HG
−1 .
(1.4)