I am learning the theory of smooth manifolds using the book Introduction to Smooth manifolds by John M. Lee, the definition of smooth subbundle in the book is as follows:
Let $\pi:E\to M$ be a smooth vector bundle, if $S\subset M$ is an embedded submanifold of $E$ whose intersection with $E_p=\pi^{-1}(p)$ is a linear subspace of $E_p$ for $\forall p\in M$ and $\pi|_S:S\to M$ is a smooth vector bundle, then $\pi|_S:S\to M$ is called a subbundle.
My question is: if $S\subset M $ is an embedded submanifold of $M$ and $S\cap E_p$ is a linear subspace of $E_p$ with the same dimension $m$ for $\forall p\in M$, is $\pi|_S:S\to M$ a subbundle? If the answer is no, could you give a counterexample, because I think this is counterintuitive. Thanks in advance!