Smooth retraction onto a differentiable manifold

Question

Let $M\subset\mathbb{R}^n$ be a smooth k-dimensional differentiable manifold (by which I mean that it is locally diffeomorphic to an open set in $\mathbb{R}^k$). Let us suppose $M$ compact for simplicity.
How can one prove that there exists an open set $U\supset M$ and a smooth retraction of $U$ onto $M$? I heard that it has to do with the smooth dependence on initial conditions of the solution of an ODE.. Can somebody shed light on this?
(Remark: clearly $U$ can't be arbitrary, since for example $\mathbb{R}^2$ does not retract onto $S^1$)

Answer 1

The important concept here is "tubular neighborhoods." These are imbeddings of a particular vector bundle (the normal bundle of M) as an open subset of the ambient manifold, and they carry the zero section to M.

In the case where the ambient manifold is not Euclidean space, the construction uses the exponential map, which may be why you heard about ODEs in connection with this. But I think that in the Euclidean case, the construction requires only some simple point-set arguments and the inverse function theorem.

You can find this construction in lots of manifold theory books. In "Topology and Geometry" by Bredon it's on page 93. I think it's also in the newest edition of Lee's "Introduction to Smooth Manifolds".