Graphics Reference
In-Depth Information
8.5
Oriented Manifolds
This section returns to the topic of orientability. Section 1.6 looked at orientation in
the context of vector spaces, which amounted to studying local orientations. In
Section 7.5 we viewed orientation in the global context of (triangulated) pseudoman-
ifolds. Now we want to describe orientation in the context of differentiable manifolds.
The new definition will be compatible with the definition for pseudomanifolds, but
will make use of the differential structure that we are assuming.
Let
M
k
be a k-dimensional submanifold of
R
n
. Since each tangent space of
M
k
is
a vector space we can talk about orientations in these tangent spaces.
Definition.
Let T :
V
Æ
W
be an isomorphism between two k-dimensional vector
spaces
V
and
W
. Define
T
*
:
orientations of
V
Æ
orientations of
W
(8.8a)
by
(
[
]
)
=
[
()( )
( )
]
T
v v
,
,...,
v
Tv
,
Tv
,...,
Tv
,
(8.8b)
*
12
k
1
2
k
where (
v
1
,
v
2
,...,
v
k
) is an ordered basis of
V
. If m is an orientation of
V
, then T
*
(m) is
called the orientation of
W
induced
by the isomorphism T.
It is easy to see that T
*
is a well-defined one-to-one correspondence between the
orientations of
V
and
W
. See Exercise 8.5.1.
Definition.
Let s be a map that associates to each
p
Œ
M
k
an orientation of the
tangent space T
p
(
M
k
). Such a choice is said to be a
continuously varying
choice of ori-
entations if for every
p
Œ
M
k
there is an open neighborhood
V
of
p
in
M
k
, a parame-
terization F :
U
Æ
V
of
V
defined on an open set
U
in
R
k
, and an orientation m of
R
k
so that DF(
q
)
*
(m) =s(F(
q
)), for
q
Œ
U
.
The definition of continuously varying orientations is simpler than it may sound.
See Figure 8.16. For example, in the case of a surface all it says is that if, say, the
tangent plane at a point of the surface has been oriented in a “counter clockwise”
fashion, then the tangent planes at nearby points are also oriented the same way. We
want to exclude random choices of orientations—some counter clockwise and others
clockwise. Note that we only chose one orientation m of
R
k
and not one in each T
q
(
R
k
)
because all those tangent spaces are the same and equal to
R
k
itself.
It is easy to show that the concept of continuously varying orientations does not
depend on any particular parameterization. See Exercise 8.5.2.
Definition.
An
orientation
of a differentiable manifold
M
is any continuously varying
choice s of orientations for the tangent spaces of
M
. A manifold is said to be
orientable
if it admits an orientation. An
oriented manifold
is a pair (
M
,s), where
M
is a mani-
fold and s is an orientation for
M
.
