Abstract
A contact twisted cubic structure({mathcal M},mathcal {C},{varvec{upgamma }}) is a 5-dimensional manifold {mathcal M} together with a contact distribution mathcal {C} and a bundle of twisted cubics {varvec{upgamma }}subset mathbb {P}(mathcal {C}) compatible with the conformal symplectic form on mathcal {C}. The simplest contact twisted cubic structure is referred to as the contact Engel structure; its symmetry group is the exceptional group mathrm {G}_2. In the present paper we equip the contact Engel structure with a smooth section sigma : {mathcal M}rightarrow {varvec{upgamma }}, which “marks” a point in each fibre {varvec{upgamma }}_x. We study the local geometry of the resulting structures ({mathcal M},mathcal {C},{varvec{upgamma }}, sigma ), which we call marked contact Engel structures. Equivalently, our study can be viewed as a study of foliations of {mathcal M} by curves whose tangent directions are everywhere contained in {varvec{upgamma }}. We provide a complete set of local invariants of marked contact Engel structures, we classify all homogeneous models with symmetry groups of dimension ge 6 up to local equivalence, and we prove an analogue of the classical Kerr theorem from Relativity.
Highlights
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
If the curvature of a given structure identically vanishes, 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
We may assume that the section σ : U → γ defining the marked contact Engel structure is of the form σ = [−t3 X1 + t2 X2 − t X3 + X4], (2.9)
Summary
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. (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. Engel’s description of the G2-invariant contact twisted cubic structure was (up to a different choice of coordinates) as follows: Let (x0, x1, x2, x3, x4) be local coordinates U ⊂ R5 and consider the coframe α0 = dx0 + x1dx4 − 3x2dx, α1 = dx, α2 = dx, α3 = dx, α4 = dx4,. 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.
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