Tubular neighborhood - Imported Knowlpedia Content

Definition

Let $i:S\hookrightarrow M$ be an embedded submanifold with normal bundle $\nu\to S$ . A tubular neighborhood of $S$ in $M$ consists of an open neighborhood $U\subseteq\nu$ of the zero section and a smooth embedding $\Phi:U\to M$ such that $\Phi(0_p)=i(p)$ for every $p\in S$ , and $\Phi(U)$ is open in $M$ . Hence $\Phi$ is a diffeomorphism from $U$ onto an open neighborhood of $S$ . Its differential in each normal fiber must induce the canonical normal direction modulo $T_pS$ .

Construction from a metric

After choosing a Riemannian metric on $M$ , the normal bundle may be represented by the orthogonal complements $T_pS^\perp$ . The Riemannian exponential map sends sufficiently small normal vectors to $M$ and restricts, with a radius allowed to vary over $S$ , to a tubular-neighborhood embedding Lee, Chapter 10. The resulting model depends on choices, although its germ has strong uniqueness properties.

Geometric uses

Fiberwise scalar multiplication in $\nu$ transports through $\Phi$ to a smooth deformation retraction of $\Phi(U)$ onto $S$ , after shrinking to a fiberwise star-shaped neighborhood if necessary. Tubular neighborhoods also make extension of sections, construction of Thom classes, and comparison of nearby submanifolds local problems on a vector bundle.

Examples and scope

For the standard inclusion $\mathbb R^k\subset\mathbb R^n$ , the normal bundle is $\mathbb R^k\times\mathbb R^{n-k}$ , and addition gives a global tubular model. A tubular neighborhood is more than an arbitrary open neighborhood of $S$ : it includes a specified identification with normal-bundle data and must agree with the inclusion along the zero section.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: Chapter 10, normal bundles and tubular neighborhoods.
Morris W. Hirsch, Differential Topology, Springer, 1976. DOI record. Relevant: Chapter 4, vector bundles and tubular neighborhoods.