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The Geometry of Marked Contact Engel Structures Gianni Manno1

· Paweł Nurowski2 · Katja Sagerschnig2

Received: 8 March 2019 / Accepted: 13 October 2020 © The Author(s) 2020

Abstract A contact twisted cubic structure (M, C, γ) is a 5-dimensional manifold M together with a contact distribution C and a bundle of twisted cubics γ ⊂ P(C) compatible with the conformal symplectic form on C. The simplest contact twisted cubic structure is referred to as the contact Engel structure; its symmetry group is the exceptional group G2 . In the present paper we equip the contact Engel structure with a smooth section σ : M → γ, which “marks” a point in each fibre γx . We study the local geometry of the resulting structures (M, C, γ, σ ), which we call marked contact Engel structures. Equivalently, our study can be viewed as a study of foliations of M by curves whose tangent directions are everywhere contained in γ. We provide a complete set of local invariants of marked contact Engel structures, we classify all homogeneous models with symmetry groups of dimension ≥ 6 up to local equivalence, and we prove an analogue of the classical Kerr theorem from Relativity. Keywords Special contact structures · Foliations · G 2 · Double fibration · Cartan’s equivalence method · Local invariants · Tanaka prolongation

1 The G2 -Geometries of Cartan and Engel In 1893 Cartan and Engel, in the same journal but independent articles [4,7], provided explicit realizations of the Lie algebra of the exceptional Lie group G2 as infinitesimal automorphisms of differential geometric structures on 5-dimensional manifolds. (In this paper G2 denotes a Lie group whose Lie algebra is the split real form of the complex

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Gianni Manno [email protected] Paweł Nurowski [email protected] Katja Sagerschnig [email protected]

1

Politecnico di Torino, Turin, Italy

2

Center for Theoretical Physics PAS, Warsaw, Poland

123

G. Manno et al.

exceptional simple Lie algebra g2 .) One of these structures was the simplest (2, 3, 5) distribution, that is, rank 2 distribution D ⊂ T N 5 on a 5-manifold N 5 such that [D, D] is a rank 3 distribution and [D, [D, D]] = T N 5 . These non-integrable distributions form an interesting and well studied (local) geometry, see Cartan’s classical paper [5] and e.g. [11] for more recent work and the associated conformal geometry. The other structure was the simplest contact twisted cubic structure. Consider a smooth 5-dimensional manifold M5 together with a contact distribution, i.e., a rank 4 subbundle C ⊂ T M5 such that the Levi bracket L : 2 C → T M5 /C, ξx ∧ ηx → [ξ, η]x modCx

(1.1)

is non-degenerate at each point x ∈ M5 . Then Lx endows each fibre Cx with the structure of a conformal symplectic vector space. Consider further a sub-bundle γ ⊂ P(C) in the projectivization of C such that each fibre γx ⊂ P(Cx ) is the image of a map RP1 → P(Cx ) ∼ = RP3 , [t, s] → [t 3 , t 2 s, ts 2 , s 3 ] ; such a curve γx is called a twisted cubic curve (or rational normal curve of degree three). Assume that the twisted cubic is Lege