---- > [!definition] Definition. ([[vertical subspace]]) > Let $E \xrightarrow{\pi} B$ be a [[vector bundle|smooth vector bundle]] over a [[smooth manifold]] $B$ with typical fiber $\mathbb{R}^{m}$. We call $\text{ker }d\pi |_{p}$ the **vertical subspace at $p \in E$**. It is a [[linear subspace]] of the [[tangent space at a point of a smooth manifold|tangent space]] $T_{p}E$. It is denoted $T_{v_{p}}E \subset T_{p}E$. We say an element of $T_{v_{p}}E$ is **vertical**. ^definition The term 'vertical' is suggestive of 'horizontal and vertical axes'. Specifically (but still perhaps a bit loosely), in a [[vector bundle|local trivialization]] $\pi ^{-1}(U)=E_{U} \xrightarrow[{\cong}]{\Phi_{U}} U \times \mathbb{R}^{m}$ $\ker d\pi |_{p}$ is the [[kernel of a linear map|kernel]] of the [[differential of a smooth map between smooth manifolds|differential]] of the coordinate projection $U \times \mathbb{R}^{m} \to U$, which is identified with the typical fiber. Then look at the picture in Lee: copies of the typical fiber are visualized vertically... This is picture that came to my mind. $U,(x^{i})_{i=1}^{\text{dim } B}$ are coordinates on $B$/local triv., $(a^{j})_{j=1}^{m=\text{dim }E}$ are coordinates on the typical fiber $\mathbb{R}^{\text{dim } E}$. So $T_{p}E=\text{span}\left( \frac{ \partial }{ \partial x^{i} }, \frac{ \partial }{ \partial y^{j} } \right)$, and $\operatorname{ker }d\pi_{p}=\text{span}\left( \frac{ \partial }{ \partial a^{j} } \right)$, which is depicted vertically in the picture. ![[Pasted image 20250506220308.png]] ---- #### ---- #### 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 > ```