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2. Submanifolds in Engel manifolds


    1. Add remark

      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. [bryant]
            • Remark. E. Murphy conjectures that non-rigid Engel knots obey an h-principle.


                • Add remark

                  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.