---- > [!definition] Definition. ([[transition functions for a vector bundle over a smooth manifold]]) > Let $E \xrightarrow{\pi} M$ be a [[smooth manifold|smooth]] [[vector bundle]] of rank $k$. Consider two trivializing neighborhoods $(U_{\alpha},\Phi_{\alpha}), (U_{\beta}, \Phi_{\beta})$ with nontrivial intersection $U_{\alpha} \cap U_{\beta}$. The map $\Phi_{\beta} \circ \Phi_{\alpha} ^{-1}$ on $(U_{\alpha} \cap U_{\beta}) \times \mathbb{R}^{k}$ behaves as $\Phi_{\beta} \circ \Phi_{\alpha}^{-1} (b, v)=(b, \psi_{\beta \alpha}(b)v)$ for some [[smooth maps between manifolds|smooth]] $\psi_{\beta \alpha}:U_{\alpha} \cap U_{\beta} \to \text{GL}_{k}(\mathbb{R})$. We call such a map $\psi_{\beta \alpha}$ a **transition function**. > Transition functions satisfy the so-called [[cocycle conditions]]. In fact, [[the Steenrod construction of a vector bundle over a smooth manifold|they determine]] the vector bundle completely. ---- #### To begin, commutativity of the diagram in the [[vector bundle]] definition enforces that $\Phi_{\beta} \circ \Phi_{\alpha}^{-1}$ fixes its first argument. Indeed, recalling that $\Phi_{\alpha}$ and $\Phi_{\beta}$ are [[homeomorphism|homeomorphisms]], and considering the diagram ```tikz \usepackage{tikz-cd} \usepackage{amsmath} \usepackage{amsfonts} \begin{document} % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB120sA9YAWgCMAXwAUAVQD6nRmgAWdAAScAxnTSKpnAEYwcdAJQhhpdJlz5CKMoKq1GLNlvayFy9mo3Pd+46ZAY2HgERILkdvTMrIggEtIuDPJKquqa8T6G7ngAtvDu2XQ4ctrawABKwjwA1sZ2MFAA5vBEoABmAE4Q2UhkIDgQSGH2UWyc3CDUDHS6DAAK5sFWIO1YDXI4fm2d3YgATNT9g9SRjjGCnDgwAB44wHA4imidAFYAdCbu3JLAzq7JHqlvHo6MJNiAOl0egcBntJlgwNEQFAIExtAxWNQ5DA6FAkGAmAwGAc6FgGGxIAiJsNThx2LM5Fh4n8stdboo6GAoIphO56YydMCwRCdr1DogRBRhEA \begin{tikzcd} \pi^{-1}(U_\alpha \cap U_\beta) \arrow[d, "\pi"'] \arrow[r, "\Phi_\alpha \text{ and } \Phi_\beta", Rightarrow] \arrow[d] & (U_\alpha \cap U_\beta) \times \mathbb{R}^k \arrow[ld, "{1\text{st proj.}, \pi_{U_\alpha \cap U_\beta}}"] \\ U_\alpha \cap U_\beta & \end{tikzcd} \end{document} ``` we see that $\begin{align} \text{1st coordinate of }\Phi_{\beta} \circ \Phi_{\alpha}^{-1} (b, v) & = \pi_{U_{\alpha} \cap U_{\beta}}\big( \Phi_{\beta} \circ \Phi_{\alpha}^{-1}(b,v) \big) \\ & = \pi( \Phi_{\alpha}^{-1}(b ,v) ) \\ & = \pi_{U_{\alpha} \cap U_{\beta}}(b, v) \\ &= b. \end{align}$ Since This leaves us with the general formula $\Phi_{\beta} \circ \Phi_{\alpha}^{-1}(b ,v)=(b, \psi_{\alpha \beta}(b ,v))$ for some $\psi_{\alpha \beta}:U_{\alpha} \cap U_{ \beta} \times \mathbb{R}^{k} \to \mathbb{R}^{k}$. Actually, $\psi_{\alpha \beta}$ is [[linear map|linear]] in its second argument, since $\Phi_{\alpha}$ and $\Phi_{\beta}$ both restrict on the fiber $E_{b}$ to [[vector space]] [[isomorphism|isomorphisms]] $E_{b} \to \{ b \} \times \mathbb{R}^{k}$ (by the v.b. definition). ---- #### 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 > ```