Foliated structure of weak nearly Sasakian manifolds
Vladimir Rovenski
TL;DR
This work extends the theory of nearly Sasakian manifolds to the broader setting of weak nearly Sasakian structures, where a skew-symmetric tensor replaces the complex structure on the contact distribution. By imposing two natural conditions, the authors show that the Reeb field is Killing and the contact distribution is curvature invariant, enabling a foliation of the manifold into two types of totally geodesic leaves. The core contribution is the eigenstructure analysis of the symmetric operator $h^2$, yielding a decomposition into distributions that define integrable, totally geodesic foliations, with Sasakian leaves arising on a subfamily; this generalizes Cappelletti-Montano–Dileo’s foliated structure results. The results offer a new framework for studying weak contact geometries and may have implications for differential geometry and mathematical physics, including twistor string theory.
Abstract
Weak contact metric manifolds, i.e., the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, defined by the author and R. Wolak, allowed a new look at the theory of contact manifolds. In this paper we study the new structure of this type, called the weak nearly Sasakian structure. We find conditions that are automatically satisfied by almost contact manifolds and under which the contact distribution is curvature invariant and the weak nearly Sasakian structure foliates into two types of totally geodesic foliations. Our main result generalizes the theorem by B. Cappelletti-Montano and G. Dileo (2016) about the foliated structure of nearly Sasakian manifolds.