Let (N;$\phi$) be a submanifold of M, such that $\phi(N)\subset M$. Let $\gamma$ : (a,b) -> M be a $C^\infty$ curve such that $\gamma(a,b)\subset \phi(N)$.
Show that it is not necessarily true that $\dot{\gamma} (t) \in T_{\gamma(t)}\phi(N)$, for each $t\in (a,b)$
Since N is a submanifold in M, $T\phi(N) \subset TM$. Therefore I assumed that the problem needs not to be solved in the choice of $\phi$. So I intrepreted the question as: find the right curve such that exist at least one $t_0 \in (a,b)$ such that $\dot{\gamma} (t_0) \notin T_{\gamma (t_0)}\phi(N)$.
Has anyone suggestions?