Geography Reference
In-Depth Information
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 ten-
sor, Euler-Lagrange deformation tensor, stretch, angular shear, areal distortion), decomposi-
tions (polar, singular value), equivalence theorems of conformal and equiareal mappings (con-
formeomorphism, areomorphism), Korn-Lichtenstein equations, optimal map projections.
There is no chance to map a curved surface (
left Riemann manifold
), which differs from a devel-
opable surface to a plane or to another curved surface (
right Riemann manifold
), without dis-
tortion 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
)ontoanother
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
.