Consider following boundary value problem of partial differential equations, \begin{align} Lu&=f, \quad \text{in} \ \Omega,\\ Bu&=0, \quad \text{on} \ \partial \Omega. \end{align} where $\Omega$ is a smooth manifold with $C^\infty$ boundary, $u \in \mathbb{R}^m$ is the unknown, $L$ is a partial differential operator, and $B=B(x) \in \mathbb{R}^{n \times m}$ is a matrix defined on $\partial \Omega.$ Moreover, the matrix $B$ is full row rank, that is $\text{rank}B=n$ and $n \leq m.$
I read a paper which considered the boundary conditions geometrically. The author said that we can take $\text{Ker}B$ as a vector bundle over $\partial \Omega.$ As a beginner at geometry, I am very confused about this statement and hope to verify it strictly according to the definition of vector bundle.
Recalling the definition (Differential Topology by Hirsch, P86), a $n$-dimensional real vector bundle $(p,E,B)$ is a continuous map from topological space $E$ to another topological space $B,$ and for every point $b \in B,$ the space $p^{-1}(b)$ is a $n$-dimensional space. Moreover, there exists an open neighborhood of $b$ and homeomorphism $\phi_U$ such that the diagram \begin{align} p^{-1}(U) \xrightarrow{\phi_U} U \times \mathbb{R}^n \\ p^{-1}(U) \xrightarrow{p} U \\ U \times \mathbb{R}^n \xrightarrow{\pi_1} U \end{align} commutes. Here $\pi_1$ is the projection to $U.$
My question is, if we take $E=\text{Ker}B$ in the above definition, how can I find the map $p$ and space $B$? Besides, in this paper, after proving that $\text{Ker}B \cong \mathbb{R}^k,$ the author said that the vector bundle $\text{Ker}B$ is also isomorphic to $\partial \Omega \times \mathbb{R}^k$, how do I understand this statement?
Thank you in advance!