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Flexibility for tangent and transverse immersions in Engel manifolds. (English) Zbl 1432.53080

An Engel structure on a smooth 4-dimensional manifold \(M\) is a maximally non-integrable distribution \(\mathcal{D}\) on \(M\). Associated to \(\mathcal{D}\) is a 3-dimensional distribution \(\mathcal{E}= [\mathcal{D},\mathcal{D}]\) and a 1-dimensional distribution \(\mathcal{W} \subset \mathcal{D}\) characterized by \([\mathcal{W}, \mathcal{E}] \subset \mathcal{E}\).
Suppose \(\mathcal{D}\) is an Engel structure on \(M\). An immersion \(\gamma : S^1 \rightarrow M\) is said to be horizontal if \(\gamma^\prime(t) \in \mathcal{D}_{\gamma(t)}\) for all \(t \in S^1\). A formal horizontal immersion is defined to be a pair of maps \(\gamma:S^1 \rightarrow M\) and \(F: S^1 \rightarrow \mathcal{D}\) such that \(0 \neq F(s) \in \mathcal{D}_{\gamma(s)}\) for all \(s\). Let \(\mathcal{HI}(\mathcal{D})\) denote the space of horizontal immersions, and let \(\mathcal{FHI}(\mathcal{D})\) denote the space of formal horizontal immersions. There is a natural inclusion \(\mathcal{HI}(\mathcal{D}) \subset \mathcal{FHI}(\mathcal{D})\) which sends \( \gamma\) to \( (\gamma, \gamma^\prime)\). The authors show that this inclusion is not always a weak homotopy equivalence because \(\mathcal{W}\) may have isolated closed orbits. On the other hand they prove that the inclusions of \(\mathcal{HI}^{\mathrm{n.e.t.}}(\mathcal{D})\), which consist of those \(\gamma \in \mathcal{HI}(\mathcal{D})\) that are not everywhere tangent to \(\mathcal{W}\), and of \(\mathcal{HI}^{\mathrm{gen}}(\mathcal{D})\), which consists of those \(\gamma \in \mathcal{HI}(\mathcal{D})\) whose \(\mathcal{W}\)-tangencies have empty interior, into \(\mathcal{FHI}(\mathcal{D})\) are both weak homotopy equivalences. In other words, an \(h\)-principle holds for \(\mathcal{HI}^{\mathrm{n.e.t.}}(\mathcal{D})\) and \(\mathcal{HI}^{\mathrm{gen}}(\mathcal{D})\).
In the last section of the paper the authors show that an \(h\)-principle also holds for immersions that are transverse to \(\mathcal{D}\).

MSC:

53C40 Global submanifolds
58A30 Vector distributions (subbundles of the tangent bundles)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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References:

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