Digital Signal Processing Reference
In-Depth Information
3.
The
2
-dimensional euclidian plane.
4.
The straight line.
The cycle
S
1
5.
.
4.5 Duality. Comparison criteria
To revisit some classical statistic geometries in low dimensions we plan to revisit the
2-dimensional complete locally flat geometry. For details see [
9
,
12
].
Owning Proposition 4.1 our interest in the 3-dimensional case is motivated by
Proposition 4.1 and Proposition 4.2. The case (1) of Proposition 4.3 agree with the
following pioneering result of Benzecri, [
9
]
Theorem 5
Up to isomorphism the only orientable complete locally flat manifold
is the flat torus
T
2
Another cruxial result we use is due to Fried et al. [
9
]. The following statement
is equivalent to their main theorem.
Theorem 6
be a compact locally flat manifold admitting a D-parallel
volume form. If its fundamental group
π
1
(
Let
(
M
,
D
)
is nilpotent then there exists a simply
connected nilpotent Lie
G
with a left invariant locally flat structure
M
)
(
G
,
D
)
and
containing a lattice
Γ
⊂
G
such that
(
M
,
D
)
=
Γ
\
(
G
,
D
).
Both Theorems 5 and 6 are used to discuss deformations of 2-dimensional statistical
models, [
5
].
Firstly, thanks to theorem of Benzecri we shall get good understanding of bundle-
like statistical structure
M
F
θ
of an optimal twisted 2-cocycles
θ
when codim
(F
θ
)
=
2
F
θ
,
D
is compact, orientable and complete.
Secondly we shall deal with simply connected non compact case.
At one side we will use the map
M
and
e
x
(
x
,
y
)
→
(
,
y
)
(
)
to identify the plane with the group
Af f
1
of affine transformations of the real
straight line.
At the other side we will use the left translation action of
Af f
in itself to
identify it with the Poincaré half plane. The Poincaré half plane has the so called
Gaussian model structure. That is an example of noncompact simply connected
statistical model.
We are going to state some claims and refer the reader to second author's PhD
dissertation [
5
].
(
1
)
