A vector field tangent to an embedded submanifold restricts to a vector field on it

In a slice chart $(U,x^1,\dots ,x^n)$ for $S$, the submanifold is given by $x^{k+1}=\cdots =x^n=0$. By [L2], write $X={\sum }_i{X}^i\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^i$ with smooth coefficients on $U$.

Tangency means that at each point of $S\cap U$ the normal components vanish: $X^{k+1}=\cdots =X^n=0$ on $S\cap U$. Hence on $S\cap U$ the field is $X{\mathrm{\mid }}_{S\cap U}={\sum }_{i=1}^k(X^i{\mathrm{\mid }}_{S\cap U})\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i},$ whose coefficients are smooth on the slice.

The local expressions from step 2.1 define a smooth section of $TS$ in each restricted slice chart, and these local sections agree on overlaps because they are all restrictions of $X$. By [L3], they therefore glue to a smooth vector field on $S$.

Therefore a smooth ambient vector field tangent to an embedded submanifold restricts to a smooth vector field on that submanifold.