I'm reading Jeffrey Lee's Manifolds and Differential Geometry. He's talking about vector fields being $\varphi$-related to each other.
He says
If $S$ is a submanifold of $M$ and $X \in \mathscr{X}(M)$, then the restriction $\left. X \right|_S \in \mathscr{X}(S)$ defined by $\left. X \right|_S(p) = X(p)$ for all $p \in S$ is $\iota$-related to $X$ where $\iota:S \hookrightarrow M$ is the inclusion map.
But why should $X(p)$ lie in $T_pS$ just because $p \in S$? For instance, let $S$ be the $xy$ plane lying in $\mathbb{R}^3$. If I take $X$ to be the constant vector field $(0,0,1)$ in $\mathscr{X}(\mathbb{R}^3)$, then I can't say that $\left. X \right|_S$ is in $\mathscr{X}(S)$, right?