Let $\psi:M\longrightarrow N$ be a $C^{\infty}$. A smooth vector field $X$ along $\psi$ (i.e. $X\in C^{\infty}(M,TN)$ $\textrm{and } \pi\circ X=\psi$) has local $C^{\infty}$ extenstions in $N$ if given $m\in M$, there exist a neighbourhood $U$ of $m$ and $V$ of $\psi(m)$, such that $\psi(U)\subset V$, and there also exist a $C^{\infty}$ vector field $\tilde{X}$ on $V$ such that $(\tilde{X}\circ\psi)|_U=X$.
I am then required to prove that a $C^{\infty}$ vector field along an immersion $\psi:M\longrightarrow N$ always has local $C^{\infty}$ exttensions in $N$.
I know that if $\psi: M^c\longrightarrow N^d$ is an immersion, and $m\in M$, then there exists a coordinate chart $(V,(x_1,x_2,..,x_d))$ in $N$ about $\psi(m)$ and an open set $U$ in $M$ about $m$, such that $\psi(U)=\{q\in V|x_i(q)=0, i=c+1,...,d\}$, i.e., $\psi(U)$ is a slice of $V$, and $\psi|_U$ is one-one.
Now I find such neighbourhoods $U$ and $V$ about $m$ and $\psi(m)$. Given a smooth vector field $X$ along $\psi$. Then $X|_{U}$ is a $C^{\infty}$ map from $U\longrightarrow TN$. How can I get $\tilde{X}$ from this?
I thought of using : If $M$ is a closed and embedded submanifold of $N$ then $C^{\infty}$ functions on $M$ are restrictions of $C^{\infty}$ functions on $N$. $\psi(U)$ is a closed submanifold of $V$, but need not be an embedded submanifold. What should I do?
Here's an outline for you:
1. On the neighborhood $V$ in your write-up above, represent $X=\sum\limits_{i=1}^c a_i(x)\dfrac{\partial}{\partial x_i}$ on $\psi(U)$, and define $\tilde X_V$ on $V$ by the same formula. (So, using these coordinates, the vector field is extended "parallel" to $\psi(U)\subset V$.)
2. Create a partition of unity $\{\rho_\alpha\}$ subordinate to an open covering $\{V_\alpha\}$ of $\psi(N)\subset M$ by such coordinate charts.
3. Define $\tilde X = \sum \rho_\alpha \tilde X_{V_\alpha}$.