From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 3

which is solved by the system of normal equations

(A − λ B)X = 0 , (1.7)

subject to

X T BX = I n . (1.8)
This is known as the general eigenvalue-eigenvector problem . The Lagrange multiplier λ is iden-
tified as eigenvalue.

End of Corollary.

Let here be given the left and right two-dimensional Riemann manifolds { M

l ,G MN } and

{ M

r ,g μν } , with standard metric G MN = G NM and g μν = g νμ , respectively, both symmetric and
positive-definite. A subset U l ⊂ M

l and U r ⊂ M r , respectively, is covered by the chart V l ⊂

E , respectively, with respect to the standard canonical

metric δ IJ and δ ij , respectively, of the left two-dimensional Euclidean space and the right two-
dimensional Euclidean space. Such a chart is constituted by local coordinates

2 :=

{ R

2 ,δ IJ }

and V r ⊂ E

2 :=

{ R

2 ,δ ij }

2

{ U, V }∈ S Ω ⊂ E

2 , respectively, over open set s S Ω and S ω .Figures 1.1 and 1.2 illustrate by
a commutative diagram the mappings

and { u,v }∈ S ω ⊂ E

Φ l maps a point from
the left two-dimensional Riemann manifold (surface) to a point of the left chart, while

Φ l , Φ r and f, f . The left mapping

Φ r maps a
point from the right t wo -dimensional Riemann manifold (surface) to a point of the right chart. In
contrast, the mapping f relates a point of the left two-dimensional Riemann manifold (surface) to a
point of the right two-dimensional Riemann manifol d ( surface). Analogously, the map pin g f maps
a point of the left chart to a point of the right chart: f :

l r , f : V l →

◦ Φ − l .All

M → M V r = Φ r ◦ f

mappings are assumed to be a diffeomorphism : the mapping is bijective.

Example 1.2 is the simple example of an isoparametric mapping of a point on an ellipsoid-of-
revolution to a point on the sphere.

{ d U, d V }→{ d u, d v }

◦ Φ − 1
l

M l → M r ; f =

Fig. 1.1. Commutative diagram ( f, f, Φ l , Φ r ); f : Φ r ◦ f