From Riemann Manifolds to Euclidean Manifolds - Map Projections: Cartographic Information Systems - page 110

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 diago-
nalization for more than two matrices.

2

Let there be given the left two-dimensional Riemann manifold { M

l ,G MN

} as well as the right two-

r ,g μν 2 ,δ μν 2 . In many applications, the choice

dimensional Euclidean manifold { M } = { R } = E

2 ,δ μν

of 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

{ R }

l of the left two-dimensional Riemann man-
ifold fixed to the point U 0 := {U 0 ,U 0 } being covered by Cartesian coordinates. (Refer to all pre-
vious 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 .Instead,we

present to you (i) the left and right eigenspace analysis and synthesis of the Cauchy-Green defor-
mation tensor, special case

{ R

2 ,δ μν }

, (ii) the left and right eigenspace analysis and
synthesis of the Euler-Lagrange deformation tensor, special case

{ M

r ,g μν }

= { R

2 ,δ μν }

r ,g μν } 2 ,δ μν }

{ M = { R . (iii) Con-

r ,g μν } 2 ,δ μν }

formeomorphism, conformal mapping, special case ; Korn-Lichtenstein

equations, special case Cauchy-Riemann equations (d'Alembert-Euler equations).

{ M = { R

2-1 Eigenspace Analysis, Cauchy-Green Deformation Tensor

Left and right eigenspace analysis and synthesis of the Cauchy-Green deformation tensor,
special case

r ,g μν } 2 ,δ μν }

{ M = { R .

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