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3.2. ORIENTATION AND THE VOLUME FORM
71
•
That is,
N
is transverse to
V
if for any point
p
∈
V
,
N
p
<
T
p
V
.
•
If
N
is transverse to
V
, then
N
,
0 on
V
.
•
Let
W
be an
n
diemsnional manifold. An (
n

1)diemsnional submanifold
M
of
W
is called a
hypersurface
in
W
.
•
A hypersurface
M
in
W
is
twosided
in
W
if there is a continuous vector
N
in
W
transverse to
M
.
•
Examples.
Normals to surfaces in
R
3
.
•
Theorem 3.2.1
A twosided hypersurface in an orientable manifold is
orientable.
•
Remarks.
•
Orientability of a manifold is an intrinsic property of a manifold.
•
Twosidedness of a manifold depends on the embedding of the manifold as
a hypersurface in a higherdimensional manifold.
•
Example.
•
Every manifold (even a nonorientable one)
M
is a twosided hypersurface
in a manifold
M
×
R
.
3.2.4
Projective Spaces
•
The real projective space
R
P
2
is the sphere
S
2
with antipodal points identi
ﬁed.
•
The sphere
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This note was uploaded on 11/26/2011 for the course MAT 4821 taught by Professor Wong during the Spring '10 term at FSU.
 Spring '10
 Wong
 Geometry

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