---- > [!definition] Definition. ([[product bundle]]) > A rank-$k$ [[vector bundle]] over any [[topological space]] $M$ is given by the [[product topology|product space]] $E=M \times \mathbb{R}^{k}$ with $\pi=\pi_{1}:M \times \mathbb{R}^{k} \to M$ as its projection. Any such bundle, called a **product bundle**, is trivial (with the [[identity map]] as a global trivialization). > If $M$ is a [[smooth manifold]], then $M \times \mathbb{R}^{k}$ is smoothly trivial. ^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 > ```