# Question about definition of map that pulls back functions defined on a domain to another domain

Let $F:M \to N$ be a diffeomorphism where $M$, $N \subset \mathbb{R}^n$ are hypersurfaces.

For $v \in L^2(N)$ let $\phi^M_N v = v \circ F$ take functions $L^2$ defined on $N$ and give out $L^2$ functions defined on $M$.

My question is how to interpret $\phi_N^M w$ if $w \in L^2(M)$. Do I take it as a constant? I know nothing else a-priori.

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Sorry, but by your definition, $\phi^M_Nw=w\circ F$, but if $w:M\to \mathbb{R}$ and $F:M\to N$, the composition $w\circ F$ isn't defined ... so what meaning do you give to $\phi^M_Nw$? In other words, $\phi^M_N$ goes from $L^2(N)$ to $L^2(M)$ ... you cannot apply it to a function in $L^2(M)$!! –  wisefool Dec 11 '12 at 21:59