Short exact sequence of vector bundles - Imported Knowlpedia Content

Definition

A short exact sequence of smooth vector bundles over a manifold $M$ is a diagram of vector bundles and bundle morphisms over $M$ ,

0\longrightarrow E' \overset{\iota}{\longrightarrow}E \overset{q}{\longrightarrow}E'' \longrightarrow0,

such that for every $x\in M$ the vector-space sequence

0\longrightarrow E'_x \overset{\iota_x}{\longrightarrow}E_x \overset{q_x}{\longrightarrow}E''_x \longrightarrow0

is exact. Equivalently, $\iota$ identifies $E'$ with the kernel bundle of $q$ , and $q$ induces a bundle isomorphism $E/\iota(E')\cong E''$ . Exactness therefore records both the embedded subbundle and its associated quotient bundle.

Immediate consequences

Fiberwise exactness implies

\operatorname{rank}E=\operatorname{rank}E'+\operatorname{rank}E''.

The morphism $\iota$ has constant rank and realizes $E'$ as a vector subbundle of $E$ ; the morphism $q$ is fiberwise surjective. Conversely, every vector subbundle $E'\subseteq E$ produces the canonical exact sequence

0\longrightarrow E'\longrightarrow E\longrightarrow E/E'\longrightarrow0.

Bundle pullback preserves short exact sequences, because taking each pulled-back fiber reproduces the original exact vector-space sequence.

Splittings

A splitting is a bundle morphism $\sigma:E''\to E$ with $q\circ\sigma=\operatorname{id}_{E''}$ . Such a choice gives an isomorphism

E'\oplus E''\longrightarrow E, \qquad (u,v)\longmapsto\iota(u)+\sigma(v).

On a paracompact smooth manifold, every short exact sequence of finite-rank smooth vector bundles splits: choose a bundle metric on $E$ and take the orthogonal complement of $\iota(E')$ . The splitting is generally noncanonical. This differs from exact sequences of holomorphic vector bundles, which need not split holomorphically.

Standard examples

For an embedded submanifold $N\subseteq M$ , the tangent-normal sequence

0\longrightarrow TN\longrightarrow TM|_N\longrightarrow \nu N\longrightarrow0

is short exact. For a constant-rank bundle morphism $\Phi:E\to F$ , its kernel and image give

0\longrightarrow\ker\Phi\longrightarrow E\longrightarrow\operatorname{im}\Phi\longrightarrow0.

These examples organize geometric information that would otherwise be recorded only as unrelated fiberwise statements.

References

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