Hi,

Is there any criteria, except for the existence of a flat connection, for a foliated bundle $E$ to be a suspension ( a foliated *flat* bundle)? For example, the Kronecker foliation on the torus is a suspension $\mathbb{R}\times_{\mathbb{Z}} \mathbb{S}^1$ , i.e. its of the form $M\times_\Gamma F$, where $\Gamma$ acts freely and transitively on the manifold $M$ and there is a free action $
\rho: \Gamma \rightarrow Diff(F)$.

Note: The Kronecker foliation is induced by the vector field , $a \frac{\partial}{\partial x} + b \frac{\partial}{\partial y}$ on $\mathbb{R}^2$ with $a,b$ constants. The action of $\mathbb{Z}$ on $\mathbb{R}\times \mathbb{S}^1$ is given by:

$(r, \exp{iz}).m= (r+m, \exp{(iz+m\alpha)})$ for some $\alpha \in \mathbb{R}$.

Thanks