The adjective "lamellar" derives from the noun "lamella", which means a thin layer.
By the Frobenius theorem this is equivalent to requiring that the Lie bracket of any smooth vector fields orthogonal to F is still orthogonal to F.[6] The condition of hypersurface-orthogonality can be rephrased in terms of the differential 1-form ω which is dual to F. The previously given Lie bracket condition can be reworked to require that the exterior derivative dω, when evaluated on any two tangent vectors which are orthogonal to F, is zero.
This can also be phrased, in terms of the Levi-Civita connection defined by the metric, as requiring that the totally anti-symmetric part of the 3-tensor field ωi∇j ωk is zero.
[9] In the special case of vector fields on three-dimensional Euclidean space, the hypersurface-orthogonal condition is equivalent to the complex lamellar condition, as seen by rewriting ω ∧ dω in terms of the Hodge star operator as ∗⟨ω, ∗dω⟩, with ∗dω being the 1-form dual to the curl vector field.
[11] In this context, hypersurface-orthogonality is sometimes called irrotationality, although this is in conflict with the standard usage in three dimensions.