---- > [!definition] Definition. ([[tautological vector bundle]]) > There is a canonical [[vector bundle]] $\gamma_{k,n}^{\mathbb{F}}$ over a [[Grassmannian]] $\text{Gr}_{k}(K)$, $K$ an $\mathbb{F}$-vector space , 'since each point is already a [[vector space]]'. Namely, one takes $\begin{align} E := \{ (V, v): V \in K, v \in V \} &\to \text{Gr}_{k}(K) \\ (V,v) &\mapsto V. \end{align}$ The fiber $E_{V}$ is manifestly $V \subset K$. >Every real [[vector bundle]] over a [[compact]] [[Hausdorff space|Hausdorff]] [[topological space|space]] can be obtained as a [[pullback of a vector bundle|pullback]] of $\gamma_{k,n}^{\mathbb{R}}$, and so one sometimes calls the vector bundles $\gamma_{-,-}^{-}$ **universal**. ^definition > [!basicexample] > - [[Real projective space|Real]] and [[complex projective space|complex]] projective spaces $\mathbb{R}P^{n}$ and $\mathbb{C}P^{n}$ come with tautological vector bundles $\gamma_{1, n+1}^{\mathbb{R}} \to \mathbb{R}P^{n}$, $\gamma_{1, n+1}^{\mathbb{C}}\to \mathbb{C}P^{n}$. Interestingly, $\gamma_{1,2}^{\mathbb{R}}$ turns out to be the [[Mobius bundle]]! ^basic-example > [!justification] > The trivializing cover may be constructed as follows (holds for the [[Grassmannian]] $\text{Gr}_{k}(\mathbb{R}^{n})$ more generally). For each line $\mathscr{L} \in \mathbb{R}P^{n}$, define the open set $U_{\ell}:=\{ \mathscr{W} \in \mathbb{R}P^{n} : \mathscr{W} \cap \mathscr{L}^{\perp}=(0)\} \subset \mathbb{R}P^{n}$. Then define $\Phi_{U_{\ell}}:\pi ^{-1}(U_{\ell}) \to U \times \ell \cong U_{\ell} \times \mathbb{R}$ as $(\mathscr{W}, w) \mapsto (\mathscr{W}, w^{\perp} )$, where $w^{\perp}$ denotes the [[orthogonal projection]] of $w$ onto $\mathscr{L}$. ^justification ---- #### ---- #### 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 > ```