The tubular neighborhood theorem November 5, 2009Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: collar neighborhood theorem, connections, exponential map, tubular neighborhood theorem
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.