Digital Signal Processing Reference
In-Depth Information
Fig. 17.1
Topological mani-
fold
U
β
ρ
1
M
φ
β
ρ
2
Intuitively, a topological manifold is a set of points which can be considered locally
as a flat Euclidean space. In other words, each point
p
∈
M
has a neighborhood
n
.Let
U
homeomorphic to an
n
-ball in
R
φ
be such an homeomorphism. The pair
(
U
, φ)
is called a coordinate neighborhood: to
p
∈
U
we assign the
n
coordinates
1
2
n
n
ξ
(
p
), ξ
(
p
), ..., ξ
(
p
)
of its image
φ(
p
)
in
R
(see Fig.
17.1
). If
p
lies also in a
1
2
n
second neighborhood
V
,let
ψ(
p
)
=[
ψ
(
p
), ρ
(
p
), ..., ρ
(
p
)
]
be its correspondent
ψ
◦
φ
−
1
n
coordinate system. The transformation
on
R
given by:
ψ
◦
φ
−
1
1
n
1
n
:[
ξ
, ..., ξ
]⇐⇒[
ρ
, ..., ρ
]
,
n
i
i
defines a local coordinate transformation on
.
In differential geometry, one is interested in intrinsic geometric properties which
are invariant with respect to the choice of the coordinate system. This can be
achieved by imposing smooth transformations between local coordinate systems
(see Fig.
17.2
). The following definition of differentiable manifold formalizes this
concept in a global setting:
Definition 17.2
R
from
φ
=[
ξ
]
to
ψ
=[
ρ
]
A differentiable (or smooth) manifold
M
is a topological manifold
with a family
U
={
U
α
, φ
α
}
of coordinate neighborhoods such that:
(1)
the
U
α
cover
M
,
(2)
for any
α, β
, if the neighborhoods intersection
U
α
∩
U
is non empty, then
β
φ
α
◦
φ
−
1
φ
β
◦
φ
−
1
and
are diffeomorphisms of the open sets
φ
β
(
U
α
∩
U
β
)
and
α
β
n
,
φ
α
(
U
α
∩
U
β
)
of
R
(3)
any coordinate neighborhood
(
V
, ψ)
meeting the property (2) with every
U
α
, φ
α
∈
U
is itself in
U
Tangent spaces
. On a differentiable manifold, an important notion (in the sequel)
is the tangent space. The tangent space
T
p
(M)
M
is the vector space of the tangent vectors to the curves passing by the point
p
.It
is intuitively the vector space obtained by a local linearization around the point
p
.
More formally, let
f
at a point
p
of the manifold
:
M
−→ R
be a differentiable function on the manifold
M
and
γ
:
I
−→
M
a curve on
M
,the
directional derivative
of f along the curve
γ
is written:
