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3.5
MANIFOLDS
We introduce now another notion which is particularly relevant in shape representation and anal-
ysis: the concept of
manifold
. is term usually identifies a space “
in which the neighborhood of each
point is just like a small piece of Euclidean space
” [
94
]. In particular,
MR
n
is a
k-dimensional
manifold
if it is locally diffeomorphic to
R
k
.
Manifolds of dimension 2 and 3 are typically the mathematical models used for 3D object
representation in geometric modeling. e manifold structure is used to tailor representation
schemes which can ensure validity of the digital models and to define operators, such as the
Euler operators, that can modify the representation while guaranteeing that at each step we still
maintain the manifold structure of the representation [
167
].
e notions underlying the theory of manifolds builds on real analysis: we refer the inter-
ested reader to the introductory topic [
118
] and the real analysis topic [
123
]. For the sake of the
mathematics described in the remainder of this chapter, we simply recall the definition of smooth
functions, also called
analytic
, which are those belonging to the class
C
1
.
Manifold without boundary
A topological Hausdorff space
M
is called a
k-dimensional topolog-
ical manifold
if each point
m2M
admits a neighborhood
U
i
M
homeomorphic to the open
disk
D
k
Dfx2R
k
jkxk< 1gR
k
and
MD
S
i2N
U
i
.
Manifold with boundary
A topological Hausdorff space
M
is called a
k-dimensional topological
manifold with boundary
if each point
m2M
admits a neighborhood
U
i
M
homeomorphic
either to the open disk
D
k
or the open half-space
R
k1
fx
n
2Rjx
n
0gR
k
and
MD
S
N
U
i
. e number
k
represents the
dimension
of the manifold.
A compact
¹
manifold without boundary is also called
closed
. Points on a manifold with
boundary are classified either
interior
, if they have a neighborhood homeomorphic to an open
disk, or
boundary
, if their neighborhood is homeomorphic to a half-disk. Figure
3.5
represents
an example of a
2
-manifold without boundary (a bitorus), a
2
-manifold with two boundaries
and a surface that in correspondence of the intersection between the plane and the bi-torus is
non-manifold.
Examples of three-dimensional manifolds with boundary are the solid sphere and the solid
torus; while their boundary, the usual sphere
S
2
and the torus
T
2
, are two closed 2-manifolds. In
the case of a terrain surface, which is usually modeled by a single-valued function, the reference
manifold
M
is a two-manifold with boundary, where all points, except those along the boundary,
have a neighborhood homeomorphic to a disk, see examples in Figure
3.6
.
i2
¹e notion of compactness generalizes the property of the subsets of the Euclidean spaces of being closed and bounded. For
a formal definition we refer to standard topics of topology, e.g., [
149
].
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