Let $X_1, X_2$ be real manifolds (endow them with Riemannian metrics if you wish).
Let $f_1:X_1 \to X_1$ continuous (or differentiable if you wish) with a robust property $A$ on an open neighborhood $\mathcal{U}_1$ of $f_1$.
Let $f_2:X_2 \to X_2$ continuous (or differentiable if you wish) with a robust property $B$ on an open neighborhood $\mathcal{U}_2$ of $f_2$.
Let finally be $f: X_1 \times X_2 \to X_1 \times X_2 / f(x,y)=(f_1(x),f_2(y))$.
I wish to know under what conditions it is possible that, given a neighborhood $\mathcal{U}$ of $f$, it can be reduced until $\mathcal{U} \subset \mathcal{U}_1 \times \mathcal{U}_2$ so that its points project onto the factors satisfying $A$ and $B$ respectively.
I don't think in general this fact to be true as a strict inclusion, but I believe this could be true in a qualitative way. My doubts come specially from the obvious fact that inside $\mathcal{U}$ there are maps that do not respect fibers so it can not be deduced striaghtforwardly. Furthermore, even when the map could respect fibers it is not straightforward that the property would hold either since conjugacies might not be easy to stablish.
Can anyone give any suggestions on the direction of the answer with recommendations of bibliographical sources? Thanks you all very much.