Engel structures

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2. Submanifolds in Engel manifolds

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Problem 2.1.
Can properties of the space of closed Engel curves be used to distinguish Engel structures? In particular for the standard Engel structure on $\mathbb{R}^4$ , is there a difference between homotopy of Engel loops and formal homotopy of Engel loops?
1. Remark. Perhaps one should exclude rigid curves from the discussion. [1]
2. Remark. E. Murphy conjectures that non-rigid Engel knots obey an h-principle.
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Problem 2.2.
Is there an h-principle for transverse embedded surfaces in a given Engel structure?
1. Remark. Already the following special case is interesting. Take a contact 3-manifold and two transverse knots that are not transversely isotopic but have equal formal invariants. Are they transverse Engel concordant through a surface in a standard Engel structure on the product of the 3-manifold with the interval?

Cite this as: AimPL: Engel structures, available at http://aimpl.org/engelstr.