Normal bundle - Knowlpedia

Let $i:S\hookrightarrow M$ be an embedded submanifold. The differential of $i$ identifies $TS$ with a vector subbundle of the restricted tangent bundle $i^*TM=TM|_S$ . The normal bundle of the embedding is the quotient vector bundle

\nu(S\subset M)=TM|_S/TS\longrightarrow S.

Its fiber at $p\in S$ is $T_pM/T_pS$ , and its rank is the codimension of $S$ in $M$ . The quotient is intrinsic to the embedding: no metric or choice of complementary subspaces is part of the definition.

Metric realization

If $M$ carries a Riemannian metric, the orthogonal complement

(TS)^\perp=\{v\in TM|_S:\langle v,w\rangle=0\text{ for all }w\in TS\}

maps isomorphically onto $\nu(S\subset M)$ . This realizes normal classes by normal vectors, but the realization depends on the metric; the quotient bundle does not.

Tubular neighborhoods

For an embedded submanifold satisfying the usual closedness hypotheses, a neighborhood of the zero section in the normal bundle is diffeomorphic to a neighborhood of $S$ in $M$ . This tubular-neighborhood theorem turns linear normal data into ambient local geometry.

Examples and non-examples

The normal bundle of the standard sphere $S^n\subset\mathbb R^{n+1}$ is a trivial line bundle, generated after choosing the Euclidean metric by the outward unit normal. An arbitrary complement of $TS$ in $TM|_S$ is a model for the normal bundle only when it varies smoothly; a pointwise choice need not form a vector subbundle.

References

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