2
1 From Riemann manifolds to Riemann manifolds
In comparing two Riemann manifolds by a mapping from one (left) to the other (right), we here only
concentrate on the corresponding metric, the first fundamental forms of two parameterized surfaces. A
comparative analysis of the second and third fundamental forms of two parameterized surfaces related
by a mapping is given elsewhere. F. Uhlig (1979) published a historical survey of the above theorem
to which we refer. Generalizations to canonically factorize two symmetric matrices A and B which
are only definite (which are needed for mappings between pseudo-Riemann manifolds) can be traced
to J. F. Cardoso and A. Souloumiac (1996), M. T. Chu (1991a,b), R. W. Newcomb (1960), C. R. Rao
and S. K. Mitra (1971), S. K. Mitra and C. R. Rao (1968), S. R. Searle (1982, pp. 312–316), W. Shougen
and Z. Shuqin (1991), and F. Uhlig (1973, 1976, 1979). In mathematical cartography, the canonical
formalism for the analysis of deformations has been introduced by N. A. Tissot (1881). Note that
there exists a beautiful variational formulation of the simultaneous diagonalization of two symmetric
matrices which motivates the notation of eigenvalues as Lagrange multipliers λ and which is expressed
by Corollary 1.2.
Corollary 1.2 (Variational formulation, 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 exist extremal (semi-)norm solutions of the Lagrange function
tr
X
T AX
1/2 =: ||X|| A , the A-weighted Frobenius norm of the non-singular matrix X subject to
the constraint
tr
X
T BX − I n
= 0 ,
(1.5)
namely the constraint optimization
||X||
2
A − λ tr
X
T BX − I n
= extr X,λ ,
(1.6)
which is solved by the system of normal equations
A − λB
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 identified
as eigenvalue.
End of Corollary.
Let here be given the left and right two-dimensional Riemann manifolds {M
2
l , G MN } and {M
2
r , g µν },
with standard metric G MN = G NM and g µν = g νµ , respectively, both symmetric and positivedefinite. A subset U l ⊂ M
2
l and U r ⊂ M
2
r , respectively, is covered by the chart V l ⊂ E
2 := {R
2 , δ IJ }
and V r ⊂ E
2 := {R
2 , δ ij }, 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 {U, V } ∈ S Ω ⊂ E
2 and {u, v} ∈ S ω ⊂ E
2 ,
respectively, over open sets S Ω and S ω . Figures 1.1 and 1.2 illustrate by a commutative diagram the
mappings Φ l , Φ r and f , f . The left mapping Φ l maps a point from the left two-dimensional Riemann
manifold (surface) to a point of the left chart, while Φ r maps a point from the right two-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
manifold (surface). Analogously, the mapping f maps a point of the left chart to a point of the right
chart: f : M
2
l → M
2
r , f : V l → V r = Φ r ◦ f ◦ Φ
−1
l . All mappings are assumed to be a diffeomorphism:
the mapping {dU, dV } → {du, dv} 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.
1 From Riemann manifolds to Riemann manifolds
In comparing two Riemann manifolds by a mapping from one (left) to the other (right), we here only
concentrate on the corresponding metric, the first fundamental forms of two parameterized surfaces. A
comparative analysis of the second and third fundamental forms of two parameterized surfaces related
by a mapping is given elsewhere. F. Uhlig (1979) published a historical survey of the above theorem
to which we refer. Generalizations to canonically factorize two symmetric matrices A and B which
are only definite (which are needed for mappings between pseudo-Riemann manifolds) can be traced
to J. F. Cardoso and A. Souloumiac (1996), M. T. Chu (1991a,b), R. W. Newcomb (1960), C. R. Rao
and S. K. Mitra (1971), S. K. Mitra and C. R. Rao (1968), S. R. Searle (1982, pp. 312–316), W. Shougen
and Z. Shuqin (1991), and F. Uhlig (1973, 1976, 1979). In mathematical cartography, the canonical
formalism for the analysis of deformations has been introduced by N. A. Tissot (1881). Note that
there exists a beautiful variational formulation of the simultaneous diagonalization of two symmetric
matrices which motivates the notation of eigenvalues as Lagrange multipliers λ and which is expressed
by Corollary 1.2.
Corollary 1.2 (Variational formulation, 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 exist extremal (semi-)norm solutions of the Lagrange function
tr
X
T AX
1/2 =: ||X|| A , the A-weighted Frobenius norm of the non-singular matrix X subject to
the constraint
tr
X
T BX − I n
= 0 ,
(1.5)
namely the constraint optimization
||X||
2
A − λ tr
X
T BX − I n
= extr X,λ ,
(1.6)
which is solved by the system of normal equations
A − λB
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 identified
as eigenvalue.
End of Corollary.
Let here be given the left and right two-dimensional Riemann manifolds {M
2
l , G MN } and {M
2
r , g µν },
with standard metric G MN = G NM and g µν = g νµ , respectively, both symmetric and positivedefinite. A subset U l ⊂ M
2
l and U r ⊂ M
2
r , respectively, is covered by the chart V l ⊂ E
2 := {R
2 , δ IJ }
and V r ⊂ E
2 := {R
2 , δ ij }, 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 {U, V } ∈ S Ω ⊂ E
2 and {u, v} ∈ S ω ⊂ E
2 ,
respectively, over open sets S Ω and S ω . Figures 1.1 and 1.2 illustrate by a commutative diagram the
mappings Φ l , Φ r and f , f . The left mapping Φ l maps a point from the left two-dimensional Riemann
manifold (surface) to a point of the left chart, while Φ r maps a point from the right two-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
manifold (surface). Analogously, the mapping f maps a point of the left chart to a point of the right
chart: f : M
2
l → M
2
r , f : V l → V r = Φ r ◦ f ◦ Φ
−1
l . All mappings are assumed to be a diffeomorphism:
the mapping {dU, dV } → {du, dv} 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.