Graphics Reference
In-Depth Information
=
(
)
FDff
,:t
MN
Æ
is called the vector bundle map of the tangent bundles
induced
by f.
8.10.6. Theorem.
(1) Df is a well-defined differentiable map from
E
(t
M
) Æ
E
(t
N
).
(2) If f :
M
Æ
M
is the identity map, then Df is the identity map on
E
(t
M
).
(2) Let f :
M
Æ
N
and g :
N
Æ
W
be differentiable maps between differentiable mani-
folds. If
hgf
=
o
:
MW
Æ
,
o
then Dh = Dg Df.
Proof.
The proof of these facts is straightforward using the corresponding proper-
ties of differentiable maps between Euclidean spaces.
Definition.
Let (
M
n
,s) and (
N
n
,t) be two oriented n-dimensional manifolds. A map
f:
M
Æ
N
is said to be
orientation preserving
or
reversing
if the induced map between
the tangent bundles is orientation preserving or reversing, respectively.
Because Theorem 8.5.3 holds for connected compact closed orientable abstract
manifolds, we can define the degree of a map f :
M
Æ
M
for such manifolds
M
just
like before.
8.10.7. Theorem.
Let
M
be a connected compact closed orientable differentiable
manifold and let f :
M
Æ
M
be a diffeomorphism. Then f is orientation preserving if
and only if deg f =+1.
Proof.
The proof involves relating what f does to the top-level homology group to
what f does to the fibers of the tangent bundle. One can do this using the definition
of the fundamental homology class in the proof of Theorem 8.5.3.
8.10.8. Example.
Let n ≥ 1. The reflection
n
n
r
:
SS
Æ
defined by
(
)
=
(
)
rxx
,...,
x
x x
,
,...,
x
12
,
n
+
1
1 2
n
+
1
is orientation reversing since its degree is -1. Because the reader may find it helpful,
we shall also work through part of the tangent bundle definition of orientability using
equations (8.15) and the coordinate neighborhood (
U
+
,p
n
), where
U
+
=
S
n
- {
e
n+1
} and
p
n
is the stereographic projection. If
p
Œ
S
n
and
p
π
e
n+1
, then equations (8.15) imply
that
