Digital Signal Processing Reference
In-Depth Information
(i) The curvature 2-form of
ω,
namely
Ω
, is defined by Maxwell equation
1
2
[
ω, ω
]
.
Ω
=
d
ω
+
(ii) A classical Theorem of Ambrose-Singer says that the holonomy algebra of
ω
is generated by the values of the curvature 2-form
Ω
. The local flatness of
D
Thus every holonomy reduction of
ω
is a covering of
M
,[
13
].
(iii) Consider a covering
π
:
Ω
=
.
implies
0
M
c
(
c
(
→
M
. Take curves
c
(
t
),
t
)
⊂
M
and
c
˜
(
t
),
˜
t
)
⊂
M
s.t.
π(
˜
c
(
c
(
c
(
c
(
c
(
t
))
=
c
(
t
),
π(
˜
t
))
=
t
),
If
c
(
0
)
=
0
)
and
c
˜
(
0
)
=˜
0
)
then
c
(
c
(
c
(
1
)
=
1
)
implies
c
˜
(
1
)
=˜
1
)
).
Now let
M
p
be the set of ends fixed homotopy classes of smooth curves
c
(
t
)
with
c
(
0
.
The following statement is classical, [
9
,
16
]
)
=
p
Theorem 2
Assume M is connected, then the following assertions are equivalent.
1.
The linear connection D is geodesically complete, (i.g every geodesic curve is
defined in the whole field of real numbers).
2.
The map
D
:M
p
→
T
p
M is one to one and onto.
Comment
.Let
c
(
t
)
be a smooth curve in a locally flat manifold
(
M
,
D
)
.For
every
v
∈
T
c
(
0
)
M
there exists a unique smooth solution
Y
(
t
)
⊂
TM
of the following
Cauchy problem
D
dc
(
t
)
dt
Y
(
t
)
=
0
,
Y
(
0
)
=
v,
0
≤
t
≤
1
.
The parallel transport
v
→
τ (v)
=
Y
(
1
)
is a linear isomorphism.
In a local coordinates functions
(
x
1
,...,
x
m
)
the Cauchy problem becomes an
system of ordinary differential equation
dx
i
dt
∂
Y
k
∂
x
i
+
dx
i
dt
=
k
ij
E
k
:
Y
j
Γ
0
,
Y
k
(
0
)
=
v
k
.
i
,
j
i
k
It is well known that the solution
Y
(
t
)
will depend smoothly on the functions
Γ
ij
.
(whose KV algebra is denoted
A
) consider
a de Rham closed 1-form
θ
s.t.
D
θ
is positive definite and consider
θ
as an element
of
C
1
Now in a locally flat manifold
(
M
,
D
)
s
be
a perturbation of
D
in the set of locally flat connections in
M
.If
s
is small enough
then
D
τ
(
A
)
. Then
(
M
,
D
θ)
is a Hessian structure in
(
M
,
D
)
.Let
D
(
s
)
=
D
+
(
s
)θ
is positive definite as well. Thus
(
M
,
D
(
s
)θ)
is a Hessian structure in
(
,
(
))
.
Now assume that
M
is compact and suppose that there is a nonexact symmetric
2-cocycle
S
M
D
s
C
2
∈
(
A
)
with
KV
(
S
)
=
0. We get the straight line
{
(
M
,
D
(
t
)),
t
∈ R}
of locally flat structures in
M
with
D
(
t
)
=
D
+
tS
.
