From Riemann Manifolds to Euclidean Manifolds - Map Projections: Cartographic Information Systems - page 126

2
l

2 ,δ μν

Box 2.6 (Canonical representation of Hilbert invariants, M →{ R } ).

I 1 (C l ):= Λ 1 + Λ 2 =tr[C l G − l ] sus i 1 (C r ):= λ 1 + λ 2 =tr[C r ] ,

(2.65)

I 2 (C l ):= Λ 1 Λ 2 =det[C l G − l ] sus i 2 (C r ):= λ 1 λ 2 =det[C r ] ,
or
I 1 (E l ):= K 1 + K 2 =tr[E l G − l ]versus i 1 (E r ):= κ 1 + κ 2 =tr[E r ] ,

(2.66)

I 2 (E l ):= K 1 K 2 =det[E l G − l ] sus i 2 (E r ):= κ 1 κ 2 =det[E r ] .
Special case: conformal mapping (syzygy).
I 1 =2 I 2 versus i 1 =2 √ i 2 . (2.67)

2 ,G MN }→{ R 2 ,δ μν }

, the first two Hilbert
invariants I 1 (E l )and i 1 (E r ) are also called left and right dilatation . They measure the isotropic
part of a deformation, while the following shear components its anisotropic part :

Note that for a general diffeomorphism, namely f : { M

Γ 1 (C l ):= C 22 − C 11 versus γ 1 (C r ):= c 22 − c 11 ,

Γ 1 (E l ):= E 22 − E 11 versus γ 1 (E r ):= e 22 − e 11 , (2.68)

Γ 2 (C l ):=2 C 12 versus γ 2 (C r ):=2 c 12 ,

Γ 2 (E l ):=2 E 12 versus γ 2 (E r ):=2 e 12 .

2-6 Polar Decomposition and Simultaneous Diagonalization of Three
Matrices

Polar decomposition and simultaneous diagonalization of three matrices: { E l , C l , G l } versus
{ E r , C r , G r } , stretch matrices.

A first remark has to be made towards the group theoretical representation of the left F l and
the right F r matrix of eigenvectors. In case of { M

2 ,δ μν } , we took advantage of the
fact that the right matrix F r of eigenvectors is an orthonormal matrix R. In the general case
{ M

r ,g μν } = { R

l ,G MN }

= { M

r ,g μν }

, the left F l and right the F r matrix of eigenvectors enjoy the polar

decomposition

F l =R 1 S 1 versus F r =R 3 S 3
versus versus , (2.69)

F l =S 2 R 2 versus F r =S 4 R 4

where the matrices R i are orthonormal, R − i =R i , while the matrices S i are by definition symmet-
ric, S i =S i . These symmetric matrices S i are sometimes called stretch matrices . or more details
including numerical examples, we refer to Marsden and Hughes ( 1983 , pp. 51-55), Ogden ( 1984 ,
pp. 92-94), Simo and Taylor ( 1991 ), and Ting ( 1985 ). Here, we conclude with a second remark