2 E. Binz - S. Pods - W. Schempp
curve β is accompanied by circular polarized waves on P
a
of arbitrarily given frequencies. This collection of periodic lifts of β defines unitary representations ρ
ν
of the Heisenberg group G
a x
, the Schr¨odinger representations cf. [11] and [13]. The frequencies of the polarized waves
correspond to the equivalence classes of ρ
ν
due to the theorem of Stone-von Neumann. The automorphism group of G
a x
is the symplectic group SpF
a x
of the symplectic complex line F
a x
. Therefore, the representation ρ
1
of G
a x
yields a projective representation of SpF
a x
, due to the theorem of Stone-von Neumann again. This projective representation is resolved to
a unitary representation W of the metaplectic group M pF
a x
in the usual way. Its infinitesimal representation d W of the Lie algebra mpF
a x
of M pF
a x
yields the quantization procedure for all homogeneous quadratic polynomials defined on the real line. Of course, this is in analogy to
the quantization procedure emanating from the quadratic approximation in optics.
2. The complex line bundle associated with a singularity free gradient field in Euclidean space
Let O be an open subset not containing the zero vector 0 in a three-dimensional oriented R- vector space E with scalar product , . The orientation on the Euclidean space E shall be
represented by the Euclidean volume form µ
E
. Our setting relies on a smooth, singularity free vector field X : O
−→ O × E with principal part a : O
−→ E, say. We shall frequently identify X with its principal part. Moreover, let H :
= R · e ⊕ E be the skew field of quaternions where e is the multiplicative unit element. The scalar product , and the orientation on E extend to all of H such
that e ∈ H is a unit vector and the above splitting of H is orthogonal. The unit sphere S
3
, i.e. Spi nE , is naturally isomorphic to SU 2 and covers S OE twice cf. [8] and [9].
Given any x ∈ O, the orthogonal complement F
a x
of ax ∈ E is a complex line as can be
seen from the following: Let C
a x
⊂ H be the orthogonal complement of F
a x
. Hence the field of quaternions H splits orthogonally into
1 H
= C
a x
⊕ F
a x
. As it is easily observed,
C
a x
= R · e ⊕ R · ax
|ax| is a commutative subfield of H naturally isomorphic to C due to
ax |ax|
2
= −e ∀ x ∈ O,
where | · | denotes the norm defined by , . This isomorphism shall be called
j
a x
: C −→ C
a x
; it maps 1 to e and i to
ax |ax|
. The multiplicative group on the unit circle of C
a x
is denoted by U
a x
1. It is a subgroup of SU 2 ⊂ H and hence a group of spins. Obviously ax generates
the Lie algebra of U
a x
1. F
a x
is a C
a x
-linear space under the right multiplication of H and hence a C-linear space, a complex line. Moreover, H is the Clifford algebra of F
a x
equipped with − , cf. [9].
The topological subspace F
a
: =
S
x ∈O
{x} × F
a x
of O × E is a C-vector subbundle of
O × E, if curl X = 0, as can easily be seen. In this case F
a
is a complex line bundle cf. [15],
Natural microstructures 3
the complex line bundle associated with X . Let pr
a
: F
a
−→ O be its projection. Accordingly there is a bundle of fields C
a
−→ O with fibre C
a x
at each x ∈ O. Clearly,
O × H = C
a
× F
a
as vector bundles over O. Of course, the bundle F
a
−→ O can be regarded as the pull-back of T S
2
via the Gauss map assigning
ax |ax|
to any x ∈ O.
We, therefore, assume that curl X = 0 from now on. Due to this assumption there is a
locally given real-valued function V , a potential of a, such that a = grad V . Each locally
given level surface S of V obviously satisfies T S = F
a
|
S
. Here F
a
|
S
= S
x ∈S
{x} × F
a x
. Each fibre F
a x
of F
a
is oriented by its Euclidean volume form i
ax |ax|
µ
E
: = µ
E ax
|ax|
, . . . , . . . . For
any level surface the scalar product yields a Riemannian metric g
S
on S given by g
S
x ; v
x
, w
x
: = v
x
, w
x
∀ x ∈ O and ∀ v
x
, w
x
∈ T
x
S. For any vector field Y on S, any x
∈ O and any v
x
∈ T
x
S, the covariant derivative ∇
S
of Levi-Civit`a determined by g
S
satisfies ∇
S v
x
Y x = dY x; v
x
+ Y x, W
a x
v
x
. Here W
a x
: T
x
S −→ T
x
S is the Weingarten map of S assigning to each w
x
∈ T
x
S the vector d
a |a|
x ; w
x
, the differential of
a |a|
at x evaluated at w
x
. The Riemannian curvature R of ∇
S
at any x is expressed by the well-known equation of Gauss as
Rx ; v
x
, w
x
. u
x
, y
x
= W
a x
w
x
, u
x
· W
a x
v
x
, y
x
2 − W
a x
v
x
, u
x
· W
a x
w
x
, y
x
for any choice of the vectors v
x
, w
x
, u
x
, y
x
∈ T
x
S. A simple but fundamental observation in our setting is that each fibre F
a x
⊂ F
a
carries a natural symplectic structure ω
a
defined by ω
a
x ; h, k := h × ax, k = h · ax, k
∀ h, k ∈ F
a x
, where
× is the cross product, here being identical with the product in H. In the context of F
a x
as a complex line we may write
ω
a
x ; h
, h
1
= |ax|· h · i, h
1
. This is due to the fact that h and ax are perpendicular elements in E . The bundle F
a
is fibre- wise oriented by
−ω
a
. In fact ω
a
extends on all of E by setting ω
a
x ; y, z := y × ax, z
for all y, z ∈ E; it is not a symplectic structure on O, of course. Let κx := det W
a x
for all x
∈ S, the Gaussian curvature of S. Provided v
x
, w
x
is an orthonormal basis of T
x
S, the relation between the Riemannian curvature R and ω is given by
Rx ; v
x
, w
x
. u
x
, y
x
= κ
x |ax|
· ω
a
x ; u
x
, y
x
for every x ∈ S and u
x
, y
x
∈ T
x
S = F
a x
.
4 E. Binz - S. Pods - W. Schempp
3. The natural principal bundle P
