Homogeneous nearly K\"ahler manifolds

TL;DR

This paper classifies all irreducible simply connected homogeneous strict nearly K"ahler manifolds, detailing their intrinsic torsion, curvature, and canonical fibrations, building on recent structural results in nearly K"ahler geometry.

Contribution

It provides an exhaustive list of such manifolds using Wolf & Gray's 3-symmetric spaces and analyzes their geometric properties in detail.

Findings

Complete classification of irreducible simply connected homogeneous strict nearly K"ahler manifolds.

Detailed descriptions of intrinsic torsion and Riemannian curvature for these manifolds.

Identification of canonical fibrations for manifolds with special algebraic torsion.

Abstract

The structure of nearly K\"ahler manifolds was studied by Gray in several papers. More recently, a relevant progress on the subject has been done by Nagy. Among other results, he proved that a strict and complete nearly K\"ahler manifold is locally a Riemannian product of homogeneous nearly K\"ahler spaces, twistor spaces over quaternionic K{\"ahler manifolds and six-dimensional nearly K\"ahler manifolds, where the homogeneous nearly K\"ahler factors are also 3-symmetric spaces. In the present paper, using the lists of $3$-symmetric spaces given by Wolf & Gray, we display the exhaustive list of irreducible simply connected homogeneous strict nearly K\"ahler manifolds. For such manifolds, we give details relative to the intrinsic torsion and the Riemannian curvature. Additionally, we determine the canonical fibration for those with special algebraic torsion.