Kernel Of surrjective linear bundle map?

Question

$E$ is a vector bundle on $M$. let $\phi:E\longrightarrow TM$ be a surjective linear bundle map. Is $\ker\phi$ vector sub-bundle of $E$?

Answer 1

The answer is yes.

Let $\phi:E\to F$ be a smooth surjective linear bundle map between two vector bundles of ranks $n+k$ and $k$ (resp.) on some manifold $M$. Then $Ker(\phi)$ has constant rank $n$, and the question is only if its locally trivial, ie can be spanned by $n$ locally defined smooth sections of $E$, pointwise linearly independent.

Fix a point $x_0\in M$ and trivialize $E$ and $F$ around $x_0$ by sections $e_1 ,\ldots,e_{n+k} $, and $f_1 ,\ldots, f_k $ (resp.), such that $e_1 , \ldots, e_n $ span $Ker(\phi)$ at $x_0$. Let $\phi(e_i)=\sum_{j=1}^k a_{ij}f_j$ for $i=1, \ldots, n$, and $\phi(e_{n+i})=\sum_{j=1}^k b_{ij}f_j$ for $i=1, \ldots, k.$ Then $A=(a_{ij})$, $B=(b_{ij})$, are $n\times k$ and $k\times k$ (resp.) matrix-valued functions defined on $M$ around $x_0$, such that $A(x_0)=0$, hence, by the surjectivity of $\phi$, $B(x_0)$ is invertible, hence $B(x)$ is also invertible near $x_0$ and $B^{-1}(x)$ is smooth.

Now let $s_i=e_i-\sum_{j=1}^k c_{ij}e_{n+j}$, $i=1,\ldots, n$, where $C=(c_{ij})=B^{-1}A.$ Then you can easily check that $s_1,\ldots, s_n$ are smooth section of $E$ that span $Ker(\phi)$ near $x_0$. QED.