Hello!


Let $E$ and $F$ be two vector bundles and let $f:E\rightarrow F$ be a bundle map. Then the kernel of $f$ is not always a subbundle of $E$. Does somebody have a simple example? Does there exist any simple conditions for which the kernel is a subbundle?


Thank you!



Kernel of a bundle map
I think this may be an example:


Consider the map from $S^1 \times \mathbb{R}^2 \rightarrow S^1 \times \mathbb{R}^2$ given by the linear map $f\_\theta: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ that takes the basis vectors $e\_1, e\_2$ to $e\_1, e\_2^\theta$, where $e\_2^\theta$ is $e\_2$ rotated by $\theta$. This should be a map of bundles over $S^1$, but the kernel isn't a subbundle as it has no trivializing neighborhood around, say, $\theta =\pi/2$. 


