Graphics Reference
In-Depth Information
Figure 8.17.
Tangent and normal vector
fields on the circle.
(a)
(b)
p
T
p
q
T
q
M
n-1
R
n
Figure 8.18.
Normal vector fields and
orientability.
With regard to the question of orientability, there is one case where normal vector
fields are especially interesting.
Let
M
n-1
be a submanifold of
R
n
. Then
M
n-1
8.5.1. Theorem.
is orientable if and
only if
M
n-1
admits a nonzero normal vector field.
Proof.
See Figure 8.18. Since the tangent space at every point of
M
n-1
is an (n - 1)-
dimensional vector subspace of
R
n
, for each point
p
of
M
n-1
we can express
R
n
uniquely as an orthogonal direct sum of the tangent space T
p
= T
p
(
M
n-1
) and a one-
dimensional subspace N
p
, that is,
n
R
=≈.
TN
p
p
Suppose now that
M
n-1
is orientable. Then
M
n-1
admits a continuously varying
choice s of orientations for its tangent spaces. Assume that s(
p
) = [
v
1
,
v
2
,...,
v
n-1
] for
some ordered basis (
v
1
,
v
2
,...,
v
n-1
) of T
p
. Choose that unique unit vector
n
p
in N
p
(there are two to choose from) so that the ordered basis (
v
1
,
v
2
,...,
v
n-1
,
n
p
) induces
the standard orientation of
R
n
. It is not hard to show that the vector field
