Graphics Reference
In-Depth Information
n
-
1
n
n:
MR
Æ
defined by
()
=
n
pn
p
(8.10)
is a nonzero normal vector field for
M
n-1
.
Conversely, suppose that
n
-
1
n
n:
MR
Æ
is a nonzero normal vector field for
M
n-1
. For each
p
Œ
M
, choose an ordered basis
(
v
1
,
v
2
,...,
v
n-1
) of T
p
so that (
v
1
,
v
2
,...,
v
n-1
,n(
p
)) induces the standard orientation of
R
n
. The map
Æ
[
]
pvv
,
,...,
n
v
(8.11)
12
-
1
will define an orientation of
M
n-1
.
It follows from Theorem 8.5.1, that we can define an orientation on an orientable
(n - 1)-dimensional manifold
M
n-1
in
R
n
simply by defining a nonzero normal vector
field. This is typically the way one does it. Conversely, if a manifold is oriented, then
the given orientation defines a unique unit normal vector field on it.
8.5.2. Example.
The normal vector field
2
3
n:
SR
Æ
defined by n(
p
) =
p
shows that the unit sphere
S
2
is orientable and defines the stan-
dard orientation of it.
Definition.
Let F :
U
Æ
M
n-1
be a regular parameterization of an (n - 1)-dimensional
submanifold
M
n-1
in
R
n
. If s is an orientation of
M
n-1
, then the unit normal
vector field of
M
n-1
described in the proof of Theorem 8.5.1 and defined by equation
(8.10) is called the
normal vector field of
M
n
-1
in
R
n
induced by
s. If s is the standard
orientation of
M
n-1
induced by F, then that normal vector field is called the
standard
normal vector field of
M
n
-1
induced by
F.
Note.
If an orientable submanifold
M
n-1
of
R
n
is closed (without boundary) and
bounded, like for example the unit sphere in
R
3
, then it divides space into bounded
and unbounded parts and it makes sense to talk about “inward” and “outward” point-
ing normals for it.
Next, we address two related natural questions. First, we know that every compact
connected differentiable manifold is a pseudomanifold (Theorem 8.3.3). Therefore, it
is reasonable to ask whether the notion of orientable defined in this section is com-
patible with that given in Section 7.5. The second question is whether there are some
simple criteria for determining the orientability of a manifold, since one certainly does
