Let $A,B$ and $C$ be three smooth manifolds. Suppose that $F:A\to C$ and $G:B\to C$ are smooth and transverse functions, making the fibered product $$S=A\underset{F,C,G}{\times}B=\{(x,y)\in A\times B\,;\,F(x)=G(y)\}$$ a manifold with tangent space at $(x,y)\in S$ $$T_{(x,y)}S=T_xA\underset{dF_x,T_zC,dG_y}{\times}T_yB$$ where $z=F(x)=G(y)$. Suppose that we have a smooth function $f:A\to\mathbb{R}$, and note $\pi_A:S\to A$ and $\pi_B:S\to B$ the canonical projections. My question is:
Is there a (generic) property we can ask $F$ to check in order to make $\tilde{f}=f\circ\pi_A:S\to\mathbb{R}$ a Morse function?
So far analysing this question we have that $d\tilde{f}:S\to T^*S$ is given by $$(x,y)\in S\mapsto \left[(\dot{a},\dot{b})\in T_{(x,y)}S\mapsto df_x(\dot{a})\right].$$ Thus $(x,y)\in S$ is critical for $\tilde{f}$ iff for all $(\dot a,\dot b)\in T_xA\times T_yB$, $dF_x(\dot a)=dG_y(\dot b)$ implies that $df_x(\dot a)=0$ (remark: if $f$ has a critical point $x_0\in A$, then $\pi_A^{-1}(x_0)$ is a set of critical points. Since this set is generically a manifold of dimension $\dim S-\dim A=\dim A+\dim B-\dim C-\dim A=\dim B-\dim C$, the function $\tilde{f}$ would be at most Morse-Bott as soon as $\dim B>\dim C$, so to avoid this case we can require $A$ to be non compact and $f$ to have no critical points). Then, $\tilde{f}$ will be a Morse function iff $d\tilde{f}$ is transverse to the zero section $0_{T^*S}\subset T^* S$. But so far I can't manage to reformulate this transversality condition as another one directly involving $F$ (I think the more likely one would be a transversality between the $2$-jet of $F$, $j^2F$, and some submanifold of the $2$-jet space). The fact that the choice of $F$ only changes implicitely $S$ and its tangent space make it difficult to me to make $F$ appear while studying the condition on $\tilde{f}$.
I will be happy to receive any idea or comment about this problem, and thank you for having read it.
