Usually, the most general condition for fiber product of manifolds (or vector bundles) to exist is that we require the images cleanly intersects. See e.g. When do fibre products of smooth manifolds ...
This question is, in a way, a follow-up of this earlier question of mine. Background Let $A$, $B$ and $F$ be finite groups and let $\alpha: A \to F$ and $\beta: B \to F$ be surjective homomorphisms. ...
Let $X$ (resp. $Y$) be the affine $k$-scheme defined by the ideal $I$ (resp. $J$) in the polynomial ring $k[x_1,...x_n]$ (resp. $k[y_1,...,y_m]$). Let $Z$ be the affine scheme defined by the ideal ...
Harold asks what conditions on $f:M\to L$ and $g:N\to L$, both smooth maps of smooth manifolds, ensures the existence of the fibre product $M \times_L N$.