Suppose $\langle\cdot, \cdot\rangle$ is a Riemannian metric on a vector bundle $(E, B, \pi)$. Let $S(E)=\{a\in E\mid \langle a,a\rangle =1\}$. Prove that $E-B$ is homotopic to $S(E)$.
I am learning the concepts of vector bundle and Riemannian metric. For the problem, I want to first show that $S(E)$ is a submanifold of $E$. Since $S(E)$ is compact, the topological embbedding part is simple. I have some difficulty of prove the submersion part and cannot figure how to go on.
Appreciate any hint or help!