AbstractA contact twisted cubic structure$$({\mathcal M},\mathcal {C},{\varvec{\upgamma }})$$
(
M
,
C
,
γ
)
is a 5-dimensional manifold $${\mathcal M}$$
M
together with a contact distribution $$\mathcal {C}$$
C
and a bundle of twisted cubics $${\varvec{\upgamma }}\subset \mathbb {P}(\mathcal {C})$$
γ
⊂
P
(
C
)
compatible with the conformal symplectic form on $$\mathcal {C}$$
C
. The simplest contact twisted cubic structure is referred to as the contact Engel structure; its symmetry group is the exceptional group $$\mathrm {G}_2$$
G
2
. In the present paper we equip the contact Engel structure with a smooth section $$\sigma : {\mathcal M}\rightarrow {\varvec{\upgamma }}$$
σ
:
M
→
γ
, which “marks” a point in each fibre $${\varvec{\upgamma }}_x$$
γ
x
. We study the local geometry of the resulting structures $$({\mathcal M},\mathcal {C},{\varvec{\upgamma }}, \sigma )$$
(
M
,
C
,
γ
,
σ
)
, which we call marked contact Engel structures. Equivalently, our study can be viewed as a study of foliations of $${\mathcal M}$$
M
by curves whose tangent directions are everywhere contained in $${\varvec{\upgamma }}$$
γ
. We provide a complete set of local invariants of marked contact Engel structures, we classify all homogeneous models with symmetry groups of dimension $$\ge 6$$
≥
6
up to local equivalence, and we prove an analogue of the classical Kerr theorem from Relativity.
Local invariants of singular surfaces in an almost complex four-manifold