Digital Signal Processing Reference
In-Depth Information
In the real projective space
RP
3
with homogeneous coordinates
(
x
,
y
,
z
,
r
)
we
consider the function
xy
2
2
F
(
x
,
y
,
z
,
r
)
=
+
z
(
z
−
r
)
We use a cellular decomposition of R P
3
to show that the projective subvariety
F
(,
x
,
y
,
z
,
r
)
=
0
is connected.
We focus on the following points.
The system
r
=
0
,
F
(
x
,
y
,
z
,
r
)
=
0
defines an irreducible projective curve in
RP
3
This projective curve is in one to one
correspondence with the variety of left (hence right) invariant complete locally flat
structures in the vector Lie group
R
2
. By the virtue of Fried-Goldman-Hirsh theorem
[
9
] our claim says that there is a bi-rational map of this projective curve on the set
of complete locally flat structures in the flat torus
2
=
R
T
2
2
.
Z
See also [
12
].
Similar ideas are used to study the non compact case
Af f
(
1
)
. Let us be more
precise.
The projective subvariety
(
,
,
,
)
=
,
=
F
x
y
z
z
0
r
0
is connected and is birationally equivalent to the set of left invariant locally flat
structures in A f f
).
To motivate our interest in
Af f
(
1
(
1
)
it may be useful to recall that topogically
Af f
(
1
)
is the 2-dimensional Gaussian model.
4.5.1 Duality in Riemannian Manifolds
This notions is attached to (pseudo) Riemannian metrics. Of course the similar con-
cept of duality symplectic versus walks. We are not dealing with it in this paper.
Let
∇
,
∇
of linear connections
(
M
, g)
be a (pseudo) Riemannian structure. A pair (
in
M
is called a dual pair in
(
M
, g)
if every triple
(
X
,
Y
,
Z
)
of smooth vector fields
satisfies the following identity
