Digital Signal Processing Reference
In-Depth Information
Indeed, let us consider the one-parameter subgroup
γ
x
(
t
)
starting from e with
initial tangent vector
x
∈ g
. As this is the integral curve of the left-invariant vector
X
=
X
field
=
DL x
, its tangent vector is
γ
x
(
t
)
=
DL
γ
x
(
t
)
x
|
γ
x
(
t
)
. The curve is a
geodesic if and only if it is auto-parallel, i.e. if
X
∇
γ
x
γ
x
=∇
X
=
α(
x
,
x
)
=
0
.
Thus, the one-parameter subgroup
γ
x
(
0.
This condition implies that the operator
α
is skew-symmetric. However, if any
skew-symmetric operator give rise to a left-invariant connection, this connection
is not always right-invariant. The connection is right-invariant if
t
)
is a geodesic if and only if
α(
x
,
x
)
=
=
DR
g
∇
X
Y
for any vector fields
X
and
Y
and any group element
g
.Aswehave
(
∇
DR
g
X
DR
g
Y
dR
g
X
X
(g
−
1
)
=
Ad
)
x
for any left-invariant vector field
=
DL x
, the right-
invariance is equivalent to the Ad-invariance of the operator
α
:
Ad
y
(g
−
1
(g
−
1
(g
−
1
α
)
,
)
=
) α(
,
),
x
Ad
Ad
x
y
for any two vectors
x
. We can focus on the infinitesimal version
of this condition by taking the derivative at
t
,
y
∈ g
and
g
∈
G
0 with
g
−
1
=
=
Exp
(
tz
)
. Since
d
dt
Ad
(
Exp
(
tz
))
x
=[
z
,
x
]
we obtain the requested characterization of bi-invariant
connections:
α(
[
.
The well known one-dimensional family of connections generated by
α(
z
,
x
]
,
y
)
+
α(
x
,
[
z
,
y
]
)
=[
z
, α(
x
,
y
)
]
x
,
y
)
=
λ
[
,
]
obviously satisfy this condition (in addition to
α(
,
)
=
0). It was shown
by Laquer [
43
] that this family describes all the bi-invariant connections on compact
simple Lie groups (the exact result is that the space of bi-invariant affine connections
on G is one-dimensional)
except
for
SU
x
y
x
x
(
n
)
when
n
>
3: in the case of
SU
(
n
)
there
is a two-dimensional family of bi-invariant affine connections.
The torsion of a connection can be expressed in the basis of left-invariant vector
fields:
T
(
X
,
Y
)
=∇
X
Y
X
−[
X
,
Y
)
− [
−∇
Y
]=
α(
x
,
y
)
−
α(
y
,
x
x
,
y
]
.
This is
itself a left-invariant vector field characterized by its value at identity
T
(
x
,
y
)
=
α(
x
,
y
)
−
α(
y
,
x
)
−[
x
,
y
]
. Thus, the torsion of a Cartan-Schouten connection is
T
(
x
,
y
)
=
2
α(
x
,
y
)
−[
x
,
y
]
and we are left with a unique torsion-free Cartan
1
connection characterized by
α(
,
)
=
2
[
,
]
x
y
x
y
.
Curvature of the Cartan-Schouten Connections
As for the torsion, the curvature tensor
R
Z
can be expressed in the basis of left-invariant vector fields and since it is left-invariant,
it is characterized by its value in the Lie algebra:
(
X
,
Y
)
Z
=∇
X
∇
Y
Z
−∇
Y
∇
X
Z
−∇
[
X
,
Y
]
R
(
x
,
y
)
z
=
α(
x
, α(
y
,
z
))
−
α(
y
, α(
x
,
z
))
−
α(
[
x
,
y
]
,
z
).
For Cartan connections of the form
α(
x
,
y
)
=
λ
[
x
,
y
]
, the curvature becomes
R
(
x
,
y
)
z
=
λ(λ
−
1
)
[[
x
,
y
]
,
z
]
.For
λ
=
0 and
λ
=
1, the curvature is obviously
