---- > [!definition] Definition. ([[orthogonal vector subbundle]]) > Let $k$ be a [[field]], let $E \xrightarrow{\pi}X$ be a [[vector bundle]] endowed with an [[inner product on a vector bundle|inner product]] $\langle -,- \rangle:E \oplus E \to k$. Given a [[vector subbundle]] $D \subset E$, define $D^{\perp}=\{ e \in E: \langle e , d \rangle=0 \text{ for all } d \text{ s.t. } \pi(d)=\pi(e) \}.$ > As one would hope from linear algebra, the natural map $D^{\perp} \hookrightarrow E \twoheadrightarrow E / D$is an [[morphism of vector bundles|isomorphism]]. This is good, because while $E / D$ can be opaque, $D^{\perp}$ is not. ---- #### [[Whitney sum of vector bundles]] ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```