Digital Signal Processing Reference
In-Depth Information
,
∇
X
Z
X
g(
Y
,
Z
)
=
g(
∇
X
Y
,
Z
)
+
g(
Y
)
,
R
∇
(
∇
,
∇
)
The curvature tensors
R
of the dual pair
are adjoint (w.r.t.
g
) each to
∇
other. In other words one has
R
∇
(
g(
R
∇
(
X
,
Y
)
Z
,
T
)
+
g(
Z
,
X
,
Y
)
T
)
=
0
.
(
∇
,
∇
)
Autodual connections are metric connections in
(
M
, g)
. A dual pair
has
∇+∇
2
opposite torsion tensors and
.Sothe
dual connection of a torsion free connection (respectively the dual of a flat connection)
is a torsion free connection (respectively is a flat connection ) as well. If
is the Levi-Civita connection of
(
M
, g)
(
M
, g,
∇
)
, g,
∇
)
is a statistical structure then
(
M
is a statistical structure as well. All of these
claims are well known, [
1
,
27
]
For every real number
α
,the
α
-connection is denoted
∇
α
. It is well known that
∇
α
,
∇
−
α
) is a dual pair.
In the vector space of linear connections in
M
any dual pair
(
(
∇
,
∇
)
and the
Levi-Civita connection of
(
M
, g)
belong to the same straight line.
, g)
contains also one
α
-connection then all of its points are
α
-connections as well.
Therefore arises the so called the local convexity problem.
Of course if a straight line
δ
containing the Levi-Civita connection of
(
M
4.5.2 Maurer-Cartan Polynomial and the Comparison Problem
Given a straight line
δ
containing the Levi-Civita connection of (M,g) what is the
cardinal of the set of locally flat dual pairs which belong to
δ
.
KV cohomology technics help to proof the following general statement:
Theorem 7
In the space of linear connections in a smooth manifold M let
LF
(δ)
be the set of points in a straightline
δ
which are locally flat. Then
1.
card
(
C
(δ)
∈ {
0
,
1
,
2
,
∞}
.
2.
If
δ
contains more than two locally flat connections then
LF
(δ)
=
δ
.
Sketch of Proof
The crux is the second assertion of our statement. Indeed let us
assume that
δ
contains three locally flat connections
D
1
,
D
2
,
D
3
such that
D
2
∈ {
ν(
t
)
=
D
1
+
t
ν,
ν
=
D
3
−
D
1
0
≤
t
≤
1
}
Let
A
be the KV algebra
(
X
(
M
),
D
1
)
and let
C
(
A
)
be its KV complex and let
C
2
C
3
PM
:
(
A
)
→
(
A
)
be its Maurer-Cartan polynomial map.
