---- > [!definition] Definition. ([[endomorphism bundle]]) > > Let $E \xrightarrow{\pi} B$ be a [[vector bundle]]. The **endomorphism bundle of $E$**, $\text{End }E$, is a [[vector bundle]] whose fiber $E_{x}$ at $x \in B$ is the homset $\text{End}(E_{p})$.[^1] [^1]: $\text{End }E$ inherits local trivializations from $E$, in the sense that any trivializing neighborhood $U \subset B$. We should be more precise here re [[the Steenrod construction of a vector bundle over a smooth manifold|Steenrod construction]]. ---- #### ---- #### 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 > ```