restricting a vector field to a leaf

Question

I have a vector field $X$ on a manifold $M$ equipped with a distribution $\cal{D}$ such that $X_p\in{\cal{D}}_p$ for all $p\in M$.

What does it mean to restrict $X$ to a leaf $L$?

Reminder: a leaf is an injective immersion tangent to the distribution. i.e. $\iota:L\rightarrow M$ is injective and $\iota_{*p}$ is injective for all $p\in M$ and $\iota_{*p}(T_pL) = {\cal{D}}_{\iota(p)}$ for all $p\in M$

The question is then what is the definition of $X\vert_L$? (ps: I need it to bee a vector field on $L$)

My idea following restricting a vector field to an open set was to define as follows

$$X\vert_L:C^\infty(L)\rightarrow C^\infty(L):f\mapsto X(f'')$$

where $f'$ is the function that $f$ defines on $\iota(L)$ and $f''\in C^\infty(M)$ which is equal to $f'$ on $\iota(L)$.

A few worries:

1. Not sure such an $f''$ exists
2. Not sure that this definition is independent of the choice of $f''$ (indeed, $\iota(L)$ is not necessarily open in $M$.

Answer 1

$X:M\rightarrow TM$.

If I can define $X^{\iota(L)} := X\vert_{\iota(L)}:\iota(L)\rightarrow TM$ and observe that actually $X_{\iota(L)} \in T_{\iota(l)}\iota(L)$, then I can define $X^L:L\rightarrow TL:p\mapsto X_{\iota(p)}$ (since $T_lL\cong T_{\iota(l)}\iota(L)$)

but $X_{\iota(l)}\in{\cal{D}}_{\iota(l)} = \iota_{*l}T_lL = T_{\iota(l)}\iota(L)$

So we have our result