Given a vector bundle $\pi \!:\! \mathcal E \rightarrow M$ a subbundle $\mathcal E^\prime$ is a subset of the total space, $\mathcal E^\prime \subset \mathcal E$ so that $$ \pi\vert_{\mathcal E^\prime}\!:\! \mathcal E^\prime \rightarrow M$$ is a vector bundle, and for each point $p\in M$ the fibre at $p$ $${\pi\vert_{\mathcal E^{\prime}}}^{-1}(p) = \mathcal E^\prime_p$$ is a vector subspace of $\mathcal E_p = \pi^{-1}(p)$