Almost complex structures and calibrated integral cycles in contact 5-manifolds

TL;DR

This paper studies special almost complex structures in 5-dimensional contact manifolds, showing that certain J-invariant integral cycles are smooth Legendrian curves and exploring their relation to calibrations.

Contribution

It establishes the smoothness of J-invariant integral cycles in contact 5-manifolds and links these structures to calibration theory, advancing understanding of contact geometry.

Findings

J-invariant integral cycles are smooth Legendrian curves

Such structures relate to calibration forms

Isolated singularities may occur in these cycles

Abstract

In a contact manifold (M^5, alpha), we consider almost complex structures J which satisfy, for any vector v in the horizontal distribution, d alpha (v,Jv) = 0. We prove that integral cycles whose approximate tangent planes have the property of being J-invariant are in fact smooth Legendrian curves except possibly at isolated points and we investigate how such structures J are related to calibrations.