# Tubular neighbourhood

A neighbourhood of a smooth submanifold $N$ in a smooth manifold $M$ that is fibred over $N$ with fibre $\mathbf R ^ {d}$, where

$$d = \mathop{\rm dim} M - \mathop{\rm dim} N.$$

Suppose that in $M$ a Riemannian metric is chosen and consider segments of geodesics that are normal to $N$ and start in $N$. If $N$ is compact, then there exists an $\epsilon > 0$ such that no two segments of length $\leq \epsilon$ and starting at different points of $N$ intersect. The union of all such segments of length $< \epsilon$ is an open neighbourhood $U$ of $N$, and is called a tubular neighbourhood of $N$. It is possible to construct for a non-compact $N$ a tubular neighbourhood by covering $N$ with a countable family of compacta and by decreasing $\epsilon$ as the number of elements of the covering increases. There is a deformation retract $r: U \rightarrow N$ associating with each point of $U$ the beginning of a geodesic containing this point. This retract determines a vector bundle with fibre $\mathbf R ^ {d}$ that is isomorphic to the normal bundle $\nu$ of the imbedding $N \rightarrow M$. In this way, the quotient space $\overline{U}\; / \partial \overline{U}\;$ is homeomorphic to the Thom space of $\nu$.

An analogue of the notion of a tubular neighbourhood can also be introduced for topological manifolds (where one has to consider locally flat imbeddings, [2]).

#### References

 [1] R. Thom, "Quelques propriétés globales des variétés différentiables" Comm. Math. Helv. , 28 (1954) pp. 17–86 [2] R.C. Kirby, L.C. Siebenmann, "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press (1977)