Can anyone give me a hint to prove the following?
If $f:M\rightarrow N$ and $g:N\rightarrow P$ are $C^{k}$ maps and $g$ is transversal to a submanifold $S$ of $P$, then: $f$ is tranversal to $g^{-1}(S)$ if and only if $g\circ f$ is transversal to $S$.
Remark: $h:A\rightarrow B$ is transversal do $W$ if $T_{h(p)}B=Dh_{p}(T_{p}A)+T_{h(p)}W$
I only found it as exercise in the books. Any clue will be great.
Thanks
Proof:
Assume $f\pitchfork g^{-1}[S]$ and $g\pitchfork S$. So you know that for every $p \in M$ we have $T_{f(p)}N = {\rm d}f_p[T_pM] + T_{f(p)}g^{-1}[S]$. Apply ${\rm d}g_{f(p)}$ on everything to get $${\rm d}g_{f(p)}[T_{f(p)}N] = {\rm d}g_{f(p)}[{\rm d}f_p[T_pM] + T_{f(p)}g^{-1}[S]] = {\rm d}(g\circ f)_p[T_pM] + {\rm d}g_{f(p)}[T_{f(p)}g^{-1}[S]]$$You also know that $T_{g(f(p))}P = {\rm d}g_{f(p)}[T_{f(p)}N] + T_{g(f(p))}S$. So add $T_{g(f(p))}S$ on both sides of the above to get $$T_{g(f(p))}P = {\rm d}(g\circ f)_p[T_pM] + {\rm d}g_{f(p)}[T_{f(p)}g^{-1}[S]] + T_{g(f(p))}S.$$However, it is easy to check that ${\rm d}g_{f(p)}[T_{f(p)}g^{-1}[S]] \subseteq T_{g(f(p))}S$. So we get that $$T_{g(f(p))}P = {\rm d}(g\circ f)_p[T_pM] + T_{g(f(p))}S,$$and so $(g\circ f)\pitchfork S$.
Assume that $(g\circ f) \pitchfork S$ and $g\pitchfork S$. Here we need to know that $g \pitchfork S$ implies that $T_{f(p)}g^{-1}[S] = ({\rm d}g_{f(p)})^{-1}[T_{g(f(p))}S]$. This is Problem 6.10 in page 148 of Lee's Introduction to Smooth Manifolds (2nd ed.). With this, take $w \in T_{f(p)}N$. So ${\rm d}g_{f(p)}(w) \in T_{g(f(p))}P$, and $(g\circ f)\pitchfork S$ gives us $v \in T_pM$ and $z \in T_{g(f(p))}S$ such that $${\rm d}g_{f(p)}(w)= {\rm d}g_{f(p)}{\rm d}f_p(v) + z,$$so that $w - {\rm d}f_p(v) \in ({\rm d}g_{f(p)})^{-1}[T_{g(f(p))}S] = T_{f(p)}g^{-1}[S]$. Then $w = {\rm d}f_p(v) + z'$ with $z' \in T_{f(p)}g^{-1}[S]$, which shows that $$T_{f(p)}N = {\rm d}f_p[T_pM]+ T_{f(p)}g^{-1}[S]$$and hence $f \pitchfork g^{-1}[S]$, concluding the proof.
Remark: one nice way to remember that in general, $F:M\to N$ and $F\pitchfork X$ for $X$ embedded in $N$ implies that $T_pF^{-1}[X]= ({\rm d}F_p)^{-1}(T_{F(p)}X)$ is using the general principle that the equation defining the tangent spaces to a submanifold is obtained by differentiating the equation defining the manifold. More precisely, the equation defining $F^{-1}(X)$ is $F(p)\in X$. "Differentiate" both sides at $p$ to describe $T_pF^{-1}[X]$ as ${\rm d}F_p(v)\in T_{F(p)}X$.