Tubular neighborhood theorem

Remarks

Such a $U$, with the diffeomorphism $\Phi$, is a tubular neighbourhood of $S$. For $M = \R^n$ the proof is direct: identify $NS$ with the bundle of vectors orthogonal to $S$, and consider $E(s, v) = s + v$. Its differential along the zero section is an isomorphism. By the inverse function theorem, $E$ is a local diffeomorphism near the zero section. An extra argument shows that $E$ is injective, hence a diffeomorphism onto an open set, on a neighbourhood of the zero section that may shrink along $S$. A fibrewise rescaling of $NS$ onto that neighbourhood finishes the proof. For a general $M$ one replaces $s + v$ by the exponential map of a Riemannian metric, or embeds $M$ in some $\R^N$. Example: the circle $S^1 \times \set{0}$ in $\R^3$ has trivial normal bundle, and a solid torus around it is a tubular neighbourhood, diffeomorphic to $S^1 \times \R^2$. A tubular neighbourhood of the central circle of the open Möbius band is the whole band, which is the total space of the nontrivial line bundle over $S^1$. Tubular neighbourhoods are unique up to isotopy.

Going deeper

For a student

The theorem is used to cut and paste manifolds along submanifolds, as in surgery, connected sums along submanifolds and Dehn surgery on knots. It identifies the quotient of $M$ by the complement of a tubular neighbourhood of $S$ with the Thom space of $NS$, which is the key step of the Pontryagin–Thom construction. It also gives a smooth retraction of a neighbourhood of $S$ onto $S$, which is used to approximate continuous maps by smooth ones. For manifolds with boundary, the companion result is the collar neighbourhood theorem.