3-quasi-Sasakian manifolds

TL;DR

This paper systematically studies 3-quasi-Sasakian manifolds, revealing their geometric structure, classification by rank, and properties of their foliations and energy minimization.

Contribution

It introduces a rank-based classification, proves a splitting theorem under certain conditions, and analyzes the foliation and energy properties of 3-quasi-Sasakian manifolds.

Findings

Reeb vector fields generate an involutive, totally geodesic foliation.

Leaves of the foliation are Lie groups: orthogonal or abelian.

Vertical distribution minimizes the corrected energy.

Abstract

In the present paper we carry on a systematic study of 3-quasi-Sasakian manifolds. In particular we prove that the three Reeb vector fields generate an involutive distribution determining a canonical totally geodesic and Riemannian foliation. Locally, the leaves of this foliation turn out to be Lie groups: either the orthogonal group or an abelian one. We show that 3-quasi-Sasakian manifolds have a well-defined rank, obtaining a rank-based classification. Furthermore, we prove a splitting theorem for these manifolds assuming the integrability of one of the almost product structures. Finally, we show that the vertical distribution is a minimum of the corrected energy.