Digital Signal Processing Reference
In-Depth Information
Π
,p
p
δX
(
t
)
X
(
t
+
dt
)
X
(
t
)
γ
(
t
+
dt
)
γ
(
t
)
T
p
Fig. 17.8
Covariant derivative along a curve
γ
DX
γ(
t
)
=∇
γ(
t
)
X
.
(17.4)
dt
Consider now two vector fields
X
and
Y
on the manifold
M
. The covariant
derivative
∇
X
Y
∈
T
p
(M)
of
Y
with respect to
X
can be defined by the following
expression:
X
i
{
∂
i
Y
k
Y
j
k
∇
X
Y
=
+
Γ
ij
}
∂
k
.
(17.5)
The expression (
17.5
) of the covariant derivative can be used as a characterization
of the connection coefficients
k
Γ
ij
. In fact, taking
X
=
∂
i
and
Y
=
∂
j
, the connection
coefficients are characterized as follows:
k
∇
∂
i
∂
j
=
Γ
ij
∂
k
.
A differentiable manifold
M
is said to be flat if and only if there exists a coordinate
k
system
[
ξ
i
]
such that the connection coefficients
{
Γ
ij
}
are identically 0. This means
that all the coordinate vector fields
.
Riemannian connection
. A Riemannian connection is an affine connection
∂
i
are parallel along any curve
γ
on
M
∇
defined on a Riemannian manifold
(M,
g
=
<, >)
such that
∀
X
,
Y
,
Z
∈
T (S)
,the
following property holds:
