4 E. Binz - S. Pods - W. Schempp
3. The natural principal bundle P
a
associated with X
We recall that the singularity free vector field X on O has the form X = id , a. Let P
a x
⊂ F
a x
be the circle centred at zero with radius |ax|
−
1 2
for any x ∈ O. Then
P
a
: =
[
x ∈O
{x} × P
a x
equipped with the topology induced by F
a
is a four-dimensional fibre-wise oriented submanifold of F
a
. It inherits its smooth fibre-wise orientation from F
a
. Moreover, P
a
is a U 1-principal bundle. U 1 acts from the right on the fibre P
a x
of P
a
via j
a x
|
U 1
: U 1 −→ U
a x
1 for any x
∈ O. This operation is fibre-wise orientation preserving. The reason for choosing the radius of P
a x
to be |ax|
−
1 2
will be made apparent below. Both F
a
and P
a
encode collections of internal variables over O and both are constructed out of X , of course. Clearly, the vector bundle F
a
is associated with P
a
. The vector field X can be reconstructed out of the smooth, fibre-wise oriented principal
bundle P
a
as follows: For each x ∈ O the fibre P
a x
is a circle in F
a x
centred at zero. The orientation of this circle yields an orientation of the orthogonal complement of F
a x
formed in E , the direction of the field at x. Hence
|ax| is determined by the radius of the circle P
a x
. Therefore, the vector field X admits a characteristic geometric object, namely the smooth, fibre-
wise oriented principal bundle P
a
on which all properties of X can be reformulated in geometric terms. Vice versa, all geometric properties of P
a
reflect characteristics of a. The fibre-wise orientation can be implemented in a more elegant way by introducing a connection form, α
a
, say, which is in fact much more powerful. This will be our next task. Since P
a
⊂ O × E, any tangent vector ξ
∈ T
v
x
P
a
can be represented as a quadruple ξ
= x, v
x
, h, ζ
v
x
∈ O × E × E × E for x
∈ O, v
x
∈ P
a x
and h, ζ
v
x
∈ E ⊂ H with the following restrictions, expressing the fact that ξ
is tangent to P
a
: Given a curve σ
= σ
1
, σ
2
on P
a
with σ
1
s ∈ O and σ
2
s ∈ P
a σ
1
s
for all s, then σ
2
s, aσ
1
s = 0 and |σ
2
s |
2
= 1
|aσ
1
s |
∀ s. Each ζ
∈ T
v
x
P
a
given by ζ =
·
σ
2
0 is expressed as ζ
= r
1
· ax
|ax| + r
2
· v
x
|v
x
| + r ·
v
x
× ax |v
x
| · |ax| with
r
1
= − W
a x
v
x
, h
, r
2
= − |v
x
| 2
· d ln |a|x; h and a free parameter r
∈ R. The Weingarten map W
a x
is of the form dax
; k = |ax| · W
a x
k + ax · d ln |a|x; k
∀ x ∈ O , ∀ k ∈ E, where we set W
a x
ax = 0 for all x ∈ O. With these preparations we define the one-form
α
a
: T P
a
−→ R
Natural microstructures 5
for each ξ ∈ T P
a
with ξ = x, v
x
, h, ζ to be
α
a
v
x
, ξ :
= v
x
× ax, ζ . 3
One easily shows that α
a
is a connection form cf. [10] and for the field theoretic aspect [1]. To match the requirement of a connection form in this metric setting, the size of the radius of P
a x
is crucial for any x ∈ O. The negative of the connection form on P
a
is in accordance with the smooth fibre-wise orientation, of course.
Thus the principal bundle P
a
together with the connection form α
a
characterizes the vector field X , and vice versa. To determine the curvature
a
which is defined to be the exterior covariant derivative of α
a
, the horizontal bundles in T P
a
will be characterized. Given v
x
∈ P
a
, the horizontal subspace H or
v
x
⊂ T P
a
is defined by H or
v
x
: = ker α
a
v
x
; . . .. A vector ξ
v
x
∈ Hor
v
x
, being orthogonal to v
x
× ax, has the form x, v
x
, h, ζ
hor
∈ O × E × E
× E where h varies in O and ζ
hor
satisfies ζ
hor
= − W
a x
v
x
, h
· ax
|ax| −
|v
x
| 2
· d ln |a|x; h · v
x
|v
x
| .
Since T pr
a
: H or
v
x
−→ T
x
O is an isomorphism for any v
x
∈ P
a
, dim H or
v
x
= 3 for all v
x
∈ P
a
and for all x ∈ O. The collection Hor ⊂ T P
a
of all horizontal subspaces in the tangent bundle T P
a
inherits a vector bundle structure T P
a
. The exterior covariant derivative d
hor
α
a
is defined by d
hor
α
a
v
x
, ξ , ξ
1
: = dα
a
v
x
; ξ
hor
, ξ
hor 1
for every ξ , ξ
1
∈ T
v
x
P
a
, v
x
∈ P
a x
and x ∈ O.
The curvature
a
: = d
hor
α
a
of α
a
is sensitive in particular to the geometry of the locally given level surfaces, as is easily verified by using equation 2:
P
ROPOSITION
1. Let X be a smooth, singularity free vector field on O with principal part a. The curvature
a
of the connection form α
a
is
a
= κ
|a| · ω
a
where κ : O −→ R is the leaf-wise defined Gaussian curvature on the foliation of O given
by the collection of all level surfaces of the locally determined potential V . The curvature
a
vanishes along field lines of X . The fact that the curvature
a
vanishes along field lines plays a crucial role in our set-up. It will allow us to establish on a simple model the relation between the transmission of internal
variables along field lines of X and the quantization of homogeneous quadratic polynomials on the real line.
4. Two examples
Kategori