---- > [!definition] Definition. ([[pullback of a vector bundle]]) > Let $E \xrightarrow{ \pi}X$ be a [[vector bundle]]. Let $f:Y \to X$ be a [[continuous|map]]. > The [[categorical pullback|pullback]] of $E$ along $f$, denoted $f^{*}E$, is instantiated by $f^{*}E=\{ (e,y) \in E \times Y : \pi(e)=f(y) \} \xrightarrow{\text{proj}_{Y}} Y.$ Note that $(f^{*}E)_{y} \cong E_{f(y)}$. > The analogue in [[sheaf|sheaf theory]] would be the [[pullback sheaf|pullback (inverse image) sheaf]]. [[restriction sheaf|Correspondingly]], we can define the **restriction of a vector bundle $E \xrightarrow{\pi} X$ to a subspace $U \subset X$** as the pullback along the [[inclusion map|inclusion]] $U \hookrightarrow X$. > The **pullback of a section $s \in \Gamma(E)$ along $f$** is $s=\big( s \circ F, \id \big)$. > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBoBGAXVJADcBDAGwFcYkQBNEAX1PU1z5CKchWp0mrdgA0efEBmx4CRUcXEMWbRCACic-kqFEy6mpqk6AZgD0AVPu7iYUAObwioKwCcIAWyQAZhocCCRiXi9fAMQyEFCkUQktdisQGkZ6ACMYRgAFAWVhEG8sVwALHAMQH38kACYQsMRyJ24gA > \begin{tikzcd} > f^*E \arrow[d] & E \arrow[d] \\ > Y \arrow[r, "f"'] & X > \end{tikzcd} > \end{document} > ``` > > [!basicproperties] Trivializations and coordinates. > Any [[vector bundle|local trivialization]] $(U, \Phi)$ of $E$ induces a [[vector bundle|local trivialization]] $(V, \Psi)$ of $f^{*}E$ by taking $V=f ^{-1} (U)$ and $\Psi(e,y):=(y, \Phi(e)_{2})$.[^2] > Suppose $X$ is moreover is a [[smooth manifold]] and $(U,\Phi)$ is moreover a [[coordinate chart]] $(x ^{i})$, with $\Phi$ inducing a [[vector bundle|local frame of sections]] $(e_{i})$ over $U$. Shrinking $U$ if necessary, we may assume $V=f^{-1} U$ is a [[coordinate chart|coordinate neighborhood]] of $Y$, $V=(y^{j})$. Then the usual coordinates $\big( (x^{i}), \Phi_{2} \big)$ on $E$ induces coordinates on $f^{*}E$ as $\big( (y^{j}), \Psi_{2} \big)=\big( (y^{j}), \Phi_{2} \big)$. ^properties > [!basicexample] > Let $M$ be a [[smooth manifold]] and $\gamma:I \to M$ a [[parameterized curve]] in $M$. Let $t$ be the [[coordinate chart|coordinate]] on $I$. > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BzOgW17oA9AFQACAKIgAvqXSZc+QigCM5KrUYs2kmXOx4CRVcvX1mrRCACy02SAz7FRMiepmtlgJLT1MKF3giUAAzACcIXiQAZmocCCQAJjdNCw5uPgEQagY6ACMYBgAFeQMlEFCsLgALHFsQ8MjEMhA46OTzNk4cGAAPHGA0cIArKQB9b2y8guLHQ0sK6trdEDCIpFUW+MQkjQ7LTjQsHykgA > \begin{tikzcd} > \gamma^* E \arrow[d, "\text{proj}_I"'] & E \arrow[d, "\pi"] \\ > I \arrow[r, "\gamma"'] & M > \end{tikzcd} > \end{document} > ``` > > We have $\gamma^{*}E= \{ (t,p): \gamma(t)= \pi (p) \}.$ > ![[Pasted image 20251104211919.png|400]] > Then coordinates for $\gamma^{*}E$ look like $(t, \Phi_{2})$ for $\Phi$ a local trivialization of $E$. [^2]: [[Well-defined]] because by definition $\pi(e)=f(y)$ so that $e \in \pi ^{-1}(f(y))$ ---- #### ---- #### 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 > ```