Complex lamellar vector field
Introduction In vector calculus, a '''complex lamellar vector field''' is a vector field which is orthogonal to a family of surfaces. In the broader context of differential geometry, complex lamellar vector fields are more often called '''hypersurface-orthogonal vector fields.''' They can be characterized in a number of different ways, many of which involve the curl. A '''lamellar vector field''' is a special case given by vector fields with zero curl.The adjective "lamellar" derives from the noun "lamella", which means a thin layer. The ''lamellae'' to which "lamellar vector field" refers are the surfaces of constant potential, or in the complex case, the surfaces orthogonal to the vector field.
Complex lamellar vector fields In vector calculus, a '''complex lamellar vector field''' is a vector field in three dimensions which is orthogonal to its own curl. That is,:The term '''lamellar vector field''' is sometimes used as a synonym for the special case of an irrotational vector field, meaning that:Complex lamellar vector fields are precisely those that are normal to a family of surfaces. An irrotational vector field is locally the gradient of a function, and is therefore orthogonal to the family of level surfaces (the equipotential surfaces). Any vector field can be decomposed as the sum of an irrotational vector field and a complex lamellar field.
Hypersurface-orthogonal vector fields In greater generality, a vector field  on a pseudo-Riemannian manifold is said to be '''hypersurface-orthogonal''' if through an arbitrary point there is a smoothly embedded hypersurface which, at all of its points, is orthogonal to the vector field. By the Frobenius theorem this is equivalent to requiring that the Lie bracket of any smooth vector fields orthogonal to  is still orthogonal to .The condition of hypersurface-orthogonality can be rephrased in terms of the differential 1-form  which is dual to . The previously given Lie bracket condition can be reworked to require that the exterior derivative , when evaluated on any two tangent vectors which are orthogonal to , is zero. This may also be phrased as the requirement that there is a smooth 1-form whose wedge product with  equals .Alternatively, this may be written as the condition that the differential 3-form  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  is zero. Using a different formulation of the Frobenius theorem, it is also equivalent to require that  is locally expressible as  for some functions  and .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  in terms of the Hodge star operator as , with  being the 1-form dual to the curl vector field.Hypersurface-orthogonal vector fields are particularly important in general relativity, where (among other reasons) the existence of a Killing vector field which is hypersurface-orthogonal is one of the requirements of a static spacetime. In this context, hypersurface-orthogonality is sometimes called ''irrotationality'', although this is in conflict with the standard usage in three dimensions. Another name is '''rotation-freeness'''.An even more general notion, in the language of Pfaffian systems, is that of a completely integrable 1-form , which amounts to the condition  as given above. In this context, there is no metric and so there is no notion of "orthogonality".
See also * Beltrami vector field* Conservative vector field
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