Let $M$ be a Riemannian smooth manifold of dimension p and $\phi$ a smooth submersion from $M$ to some other smooth manifold $N$ of dimension q. Denote by $\mathcal{F}$ the foliation of leaves $\phi^{-1}(z)$ for $z\in \phi(M)$. When can we say that $M$ is isomorphic to a product manifold of the form $L\times S$ where $L$ is a leave of $\mathcal{F}$ passing through a point $x_0$ and $S$ is a transversal leave passing through the same point $x_0$.
A classical condition in ([1], Theorem 4.4 section 4.4) says that if $M$ is complete simply connected and if $\mathcal{F}$ is parallel then $M$ is isomorphic to such product manifold. But I don't know what condition $\phi$ should satisfy to ensure that $\mathcal{F}$ is parallel?
Are there more general conditions on $\phi$ to ensure that $M$ can be seen as a product manifold?
[1] A. Bejancu and H. R. Farran. Foliations and Geometric Structures. Springer Science & Business Media, Jan. 2006. Google-Books-ID: NtqEFj4nYecC.