Foliated bundles and suspensions

Question

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

Answer 1

I am interpreting the terminology as follows:

• foliated bundle = foliation on the total space of the bundle
• foliated flat bundle = suspension

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If the fibers of the bundle are compact then the only thing you need is transversality between leaves and fibers.

If the fibers are not compact then besides transversality you also need to prevent the leaves from scaping to infinity. To do it, you can ask the natural projections from leaves to the base of your bundle to be covering maps.

See Chapter V of Geometric Theory of Foliations.