30
1 From Riemann manifolds to Riemann manifolds
I n order to visualiz e the eigenspace of both the left and the right Euler–Lagrange deformation tensor
E l and E r relative to the left and right metric tensors G l and G r , we are forced to compute in addition
the left and right eigenvectors (namely the left and right eigencolumns, also called eigendirectories)
of the pairs {E l , G l } and {E r , G r }, respectively. Lemma 1.8summariz es the results.
Lemma 1.8 (Left and right eigenvectors of the left and the right Euler–Lagrange deformation tensor).
For the pair of symmetric matrices {E l , G l } or {E r , G r }, an ex plicit form of the left eigencolumns and
the right eigencolumns is
1st left eigencolumns, K 1 :
F 11
F 21
=
1
√
G 11 (e 22 −K 1 G 22 ) 2 −2G 12 (e 12 −K 1 G 12 )(e 22 −K 1 G 22 )+G 22 (e 12 −K 1 G 12 ) 2
×
×
e 22 − K 1 G 22
−(e 12 − K 1 G 12 )
;
(1.115)
2nd left eigencolumns, K 2 :
F 12
F 22
=
1
√
G 22 (e 11 −K 2 G 11 ) 2 −2G 12 (e 11 −K 2 G 11 )(e 12 −K 2 G 12 )+G 11 (e 12 −K 2 G 12 ) 2
×
×
−(e 12 − K 2 G 12 )
e 11 − K 2 G 11
;
(1.116 )
1st right eigencolumns, κ 1 :
f 11
f 21
=
1
√
g 11 (E 22 −κ 1 g 22 ) 2 −2g 12 (E 12 −κ 1 g 12 )(E 22 −κ 1 g 22 )+g 22 (E 12 −κ 1 g 12 ) 2
×
×
E 22 − κ 1 g 22
−(E 12 − κ 1 g 12 )
;
(1.117 )
2nd right eigencolumns, κ 2 :
f 12
f 22
=
1
√
g 22 (E 11 −κ 2 g 11 ) 2 −2g 12 (E 11 −κ 2 g 11 )(E 12 −κ 2 g 12 )+g 11 (E 12 −κ 2 g 12 ) 2
×
×
−(E 12 − κ 2 g 12 )
E 11 − κ 2 g 11
.
(1.118 )
End of Lemma.
The proof of these relations follows the line of thought of the proof of Lemma 1.6 . Accordingly, we
sk ip any proof here.