18
1 From Riemann manifolds to Riemann manifolds
∂/∂U
–
dV
1
∂/∂V
–
dV
2
∂/∂U
–
dV
1
∂/∂V
–
dV
2
Λ 1
Λ 2
Fig. 1.7. Left Cauchy–Green tensor, left Tissot circle S
1 , left Tissot ellipse E
1
Λ 1 ,Λ 2 , the tangent vectors are
∂/∂U and ∂/∂V .
The canonical forms of the metric, namely dS
2 and ds
2 , have been interpreted as the following pairs:
left Tissot circle S
1
versus
left Tissot ellipse E
1
Λ 1 ,Λ 2
,
and
right Tissot ellipse E
1
λ 1 ,λ 2
versus
right Tissot circle S
1 .
(1.70)
Figure 1.7 illustrates the pair {left Cauchy–Green deformation tensor, left metric tensor} by means of
the left Tissot circle S
1 and the left Tissot ellipse E
1
Λ 1 ,Λ 2
on the left tangent space T M
2
l . In contrast,
by means of Fig. 1.8 , we aim at illustrating the pair {right Cauchy–Green deformation tensor, right
metric tensor} by means of the right Tissot ellipse E
1
λ 1 ,λ 2
and the right Tissot circle S
1 on the right
tangent space T M
2
r . The left eigenvectors span canonically the left tangent space T M
2
l , while the right
eigenvectors span the right tangent space T M
2
r , namely
U
M
A
∂
∂U M versus u
µ
α
∂
∂u µ ,
F l
∂
∂U
versus F r
∂
∂u
.
(1.71)
Indeed, they are generated from a dual holonomic base (coordinate base) {dU
1 , dU
2
} versus {du
1 , du
2
}
to an anholonomic base { –
dV
1 , –
dV
2
} = {Ω 1 , Ω 2 } versus { –
dv
1 , –
dv
2
} = {ω 1 , ω 2 } by the transformations
dU
1
dU
2
= F l
Ω 1
Ω 2
versus
du
1
du
2
= F r
ω 1
ω 2
.
(1.72)
∂/∂u
–
dv
1
∂/∂v
–
dv
2
∂/∂u
–
dv
1
∂/∂v
–
dv
2
λ 1
λ 2
Fig. 1.8. Right Cauchy–Green tensor, right Tissot ellipse E
1
λ 1 ,λ 2 , right Tissot circle S
1 , the tangent vectors
are ∂/∂u and ∂/∂v.