The Geometry of Marked Contact Engel Structures

The Journal of Geometric Analysis https://doi.org/10.1007/s12220-020-00545-5 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 B Gianni Manno Paweł Nurowski Katja Sagerschnig 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 Legendrian, which means that it is compatible with the conformal symplectic structure on the contact plane (see Sect. 2.2 for details). Then (M5 , C, γ) is called a contact twisted cubic structure. Both geometries, (2, 3, 5) distributions as well as contact twisted cubic structures, are examples of parabolic geometries, see [6]. As such, they admit canonical Cartan connections, whose curvature gives rise to the fundamental invariants of these structures. If the curvature of a given structure identically vanishes, then the structure is locally equivalent to the flat model of the geometry under consideration: In case of a (2, 3, 5) distribution this is the G2 -invariant (2, 3, 5) distribution on the flag manifold G2 /P1 and in case of a contact twisted cubic structure this is the G2 -invariant contact twisted cubic structure on the flag manifold G2 /P2 . Here we use the standard notation P1 and P2 for the two 9-dimensional maximal parabolic subgroups of G2 . The geometric structures presented by Cartan and Engel are local coordinate description of the two flat models. Engel’s description of the G2 -invariant contact twisted cubic structure was (up to a different choice of coordinates) as follows: Let (x 0 , x 1 , x 2 , x 3 , x 4 ) be local coordinates U ⊂ R5 and consider the coframe α 0 = dx 0 + x 1 dx 4 − 3x 2 dx 3 , α 1 = dx 1 , α 2 = dx 2 , α 3 = dx 3 , α 4 = dx 4 , (1.2) with dual frame X 0 = ∂x 0 , X 1 = ∂x 1 , X 2 = ∂x 2 , X 3 = 3x 2 ∂x 0 + ∂x 3 , X 4 = −x 1 ∂x 0 + ∂x 4 . (1.3) Here α 0 is a contact form and defines a contact distribution C = ker(α 0 ). Now consider the set of horizontal null vectors 123 The Geometry of Marked Contact Engel Structures γ̂ = { Y ∈ C : g1 (Y , Y ) = g2 (Y , Y ) = g3 (Y , Y ) = 0 } of the three degenerate metrics g1 = α 1 α 3 − (α 2 )2 , g2 = α 2 α 4 − (α 3 )2 , g3 = α 2 α 3 − α 1 α 4 , (1.4) where α i α j = 21 (α i ⊗ α j + α j ⊗ α i ). Then Y ∈ (C) takes values in γ̂ if and only if is of the form Y = t 3 X 1 + t 2 s X 2 + ts 2 X 3 + s 3 X 4 . Hence the projectivization γx ⊂ P(Cx ) of γ̂x is a twisted cubic curve, and it is straightforward to verify that it is Legendrian. A contact twisted cubic structure that is locally equivalent to the G2 -invariant structure (U, C, γ) described above will be called a contact Engel structure.1 2 Marked Contact Engel Structures and a Kerr Theorem On a contact Engel structure there is, at each point x ∈ M5 , a distinguished set of directions, namely those corresponding to points p ∈ γx . In this work, we equip the contact Engel structure (possibly after restricting to an open subset of M5 ) with a section σ that marks a point ∗ = σ (x) in each twisted cubic γx . Definition 1 A marked contact Engel structure (U, C, γ, σ ) is a contact Engel structure together with a smooth section σ : U → γ ⊂ P(C) of the bundle RP1 → γ → U of twisted cubics. Since γx ⊂ P(Cx ) is cut out by the three polynomials (1.4) and because of the analogy with Lorentzian geometry to be discussed below, we refer to directions in γ as null directions.2 A marked contact Engel structure can be thought of as a null congruence structure, that is, a (local) foliation of the contact Engel structure by horizontal null curves. For each x ∈ U, the point σ (x) ∈ γx corresponds to a null direction σx in the contact plane Cx . Therefore the section σ defines a rank one distribution σ ⊂ T U whose integral curves define the null congruence. 2.1 Analogy with Null Congruence Structures in Lorentzian Geometry Conformal Lorentzian geometries (M4 , [g]) in 4-dimensions are the geometries studied in General Relativity when the related physics is concerned with massless particles 1 Contact Engel structures should not be confused with Engel distributions, sometimes also called Engel structures, which are maximally non-integrable rank 2 distributions on 4-dimensional manifolds. 2 The analogy is even more striking if (...truncated)