Quotient vector bundle

Let $E\to M$ be a smooth vector bundle and let $F\subseteq E$ be a smooth vector subbundle. The quotient vector bundle $E/F\to M$ has fiber

(E/F)_x=E_x/F_x.

Its total space is the quotient of $E$ by the equivalence relation $v\sim w$ when $v,w\in E_x$ and $v-w\in F_x$ . Local bundle frames adapted to $F$ identify $E/F$ with $U\times\mathbb F^{r-k}$ , and these charts give it a unique smooth vector-bundle structure for which the canonical map $q:E\to E/F$ is a smooth, fiberwise-surjective vector bundle morphism with kernel $F$ .

Local construction

Choose a local frame $e_1,\ldots,e_r$ of $E$ such that $e_1,\ldots,e_k$ frame $F$ . Then the residue classes of $e_{k+1},\ldots,e_r$ form a local frame of $E/F$ . Consequently,

\operatorname{rank}(E/F)=\operatorname{rank}(E)-\operatorname{rank}(F).

Changes between adapted frames descend to invertible changes of quotient frames, so the construction is independent of the chosen frames.

Exact sequence and splittings

The quotient projection fits into the short exact sequence

0\longrightarrow F\longrightarrow E\overset{q}{\longrightarrow}E/F\longrightarrow0.

After choosing a bundle metric, the orthogonal complement $F^\perp$ maps isomorphically onto $E/F$ . This gives a smooth splitting on the usual paracompact smooth-manifold bases, but it depends on the metric and is not canonical. The quotient itself requires no such choice.

Functoriality and examples

For a smooth map $f:N\to M$ , pullback preserves the quotient construction:

f^*(E/F)\cong f^*E/f^*F.

If $N\subseteq M$ is an embedded submanifold, its normal bundle is the quotient $TM|_N/TN$ . More generally, the cokernel of an injective constant-rank bundle morphism is a quotient vector bundle.

References

D. Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: Chapter 3, subbundles, quotient bundles, and exact sequences.
L. W. Tu, Differential Geometry: Connections, Curvature, and Characteristic Classes, Springer, 2017. DOI record. Relevant: Chapter 1, operations on vector bundles.