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Engel structures and weakly hyperbolic flows on four-manifolds. (English) Zbl 1402.37045

An {Engel structure} on a \(4\)-dimensional manifold \(M\) is a smooth rank-\(2\) distribution \(\mathcal{D}\) with the property that \([\mathcal{D}\),\( \mathcal{D]}\) is an even contact structure \(\mathcal{E}\) (i.e., a maximally non-integrable smooth hyperplane field \(\mathcal{E}\)).
A { bi-Engel structure} on a \(4\)-dimensional manifold \(M\) is a pair of Engel structures \((\mathcal{D}_{+}\),\(\mathcal{D}_{-}\mathcal{)}\) inducing the same even contact structure \(\mathcal{E}\), defining opposite orientations for \(\mathcal{E}\) and having a one-dimensional intersection.
In the paper under review, the authors study bi-Engel structures. The main result of the paper gives the correspondence between bi-Engel structures and a class of weakly hyperbolic flows.

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
58A30 Vector distributions (subbundles of the tangent bundles)
37D30 Partially hyperbolic systems and dominated splittings
53D35 Global theory of symplectic and contact manifolds
53D10 Contact manifolds (general theory)
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