Direct sum vector bundle (Whitney sum)

Let $\pi_E:E\to M$ and $\pi_F:F\to M$ be smooth vector bundles over the same smooth manifold $M$ . Their direct sum bundle (or Whitney sum) is the vector bundle

\pi_{E\oplus F}:E\oplus F \to M

defined by

E\oplus F := \{(e,f)\in E\times F : \pi_E(e)=\pi_F(f)\}, \qquad \pi_{E\oplus F}(e,f):=\pi_E(e)=\pi_F(f),

with fiberwise vector space structure

(E\oplus F)_x \cong E_x\oplus F_x.

Local trivializations of $E$ and $F$ over an open set $U$ induce a local trivialization of $E\oplus F$ over $U$ by taking the product trivialization and using the usual direct sum on $\mathbb F^{r}\oplus\mathbb F^{s}$ .

The projections $\mathrm{pr}_E:E\oplus F\to E$ and $\mathrm{pr}_F:E\oplus F\to F$ are vector bundle morphisms over $\mathrm{id}_M$ .

The rank satisfies $\mathrm{rank}(E\oplus F)=\mathrm{rank}(E)+\mathrm{rank}(F)$ (see rank of a vector bundle).

Examples

Tangent plus cotangent. $TM\oplus T^*M$ is a bundle whose fiber at $x$ is $T_xM\oplus T_x^*M$ ; it is fundamental in generalized geometry.

Trivial sums. If $E\cong M\times \mathbb F^r$ and $F\cong M\times \mathbb F^s$ , then $E\oplus F\cong M\times \mathbb F^{r+s}$ .

Splitting by a metric. If $E$ has a bundle metric and $S\subset E$ is a subbundle, then (under standard hypotheses) $E$ can be identified with $S\oplus S^\perp$ by mapping $(s,t)$ to $s+t$ fiberwise.