If is a manifold and
a compact submanifold, then a tubular neighborhood of
consists of an open set
diffeomorphic to a neighborhood of the zero section in some vector bundle
over
, by which
corresponds to the zero section.
Theorem 1 Hypotheses as above,
has a tubular neighborhood.
When , the idea is to take
as the normal bundle of
, whose fiber at
consists of tangent vectors in
perpendicular to
. There is a map
sending
for
to
. When restricted to the zero section, this map is just the identity on
, and the differential is the natural map
which is an isomorphism, by construction. Using the inverse function theorem, we can find a bounded neighborhood containing the zero section, such that
is locally a homeomorphism.
Now, following Spivak, I quote a lemma:
Lemma 2 If
is a closed subset of the compact metric space
, and
is a local homeomorphism that is injective on
, then there is a neighborhood
with
injective.
Suppose this failed; then we can find a sequence of open sets containing
(e.g. the
neighborhood) with
such that
isn’t one-to-one on
. Thus one picks
with
. By taking subsequences if necessary, assume both
converge to
. Then clearly
. I claim
; otherwise for
very large there would be
close to
with
, and this contradicts the local homeomorphism condition. Now
but
, so this contradicts the injectivity on
.
So returning to the special case of the theorem, we can find a suitably small neighborhood such that the map
is injective, hence an isomorphism onto the image.
Now for the general case. We can make into a metric space via a Riemannian metric, which can be constructed through a standard partition-of-unity argument. Similarly, we can obtain a connection on
(not necessary for it to be compatible with the Riemannian metric), which leads to exponential maps as before.
We can restrict the bundle to
to get a vector bundle
, which contains as a subbundle
. Let
be a complement (e.g. obtained by putting a nondegenerate inner product on
) of
, i.e. such that
.
Now there is an open neighborhood of the zero section in
and the exponential map
; this is smooth by the ODE theorem, and the differential at
for
is seen to be an isomorphism as before (yesterday we computed the differential of the exponential map at
). This fact, with the lemma above, completes the proof of the tubular neighborhood theorem.
Note in particular that is a smooth deformation retract of the tubular neighborhood
(via
,
).
Here is another variant of this result:
Theorem 3 (Collar Neighborhood Theorem) Let
be a manifold-with-boundary, with compact boundary
. There is a neighborhood
isomorphic to
for
an open interval.
Choose a Riemannian metric on (which is defined in the same way as for a regular manifold). On each point
of the boundary
, we can assign a normal
perpendicular to
that points “inward” to
. We can find a connection
on
, and using the exponential map, we can consider
. Then
is defined on
for
small enough. Now this is a local isomorphism at each point of
, and we can apply the same type of argument as before.
Leave a Reply