---- > [!definition] Definition. ([[horizontal subspace]]) > Let $E \xrightarrow{\pi} B$ be a [[vector bundle|smooth vector bundle]] over a [[smooth manifold]] $B$. Let $T_{v_{p}}E$ be the [[vertical subspace]] at $p \in E$. We call any [[complement of a linear subspace|complementary subspace]] $S_{p} \subset T_{p}E$ to $T_{v_{p}}E$ a **horizontal subspace at $p$**. > > It has dimension $\dim B$.[^1] ^definition > [!intuition] > The naming of vertical and horizontal subspaces are suggestive of vertical and horizontal coordinate axes when graphing a function $\mathbb{R} \to \mathbb{R}$. ^intuition [^1]: This is e.g. because $\pi$ being a [[smooth submersion]] implies $\dim \im d\pi |_{p}=\dim B$, hence $\dim \ker d\pi |_{p}=\dim E - \dim B$ by [[Rank-Nullity theorem|rank-nullity]], hence $\dim S_{p}=\dim E - \dim T_{v_{p}}E=\dim B$ by [[direct sum of modules|direct sum]] and [[dimension]] considerations. Equivalently, observe from the [[first isomorphism theorem for modules]] that $S_p \cong T_{p}E / T_{v_{p}}E$ and subtract dimensions. ---- #### ---- #### 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 > ```