Nearly Kaehler geometry and Riemannian foliations

TL;DR

This paper investigates the structure of strict, complete nearly Kaehler manifolds with the canonical Hermitian connection, revealing their local product structure and classifying homogeneous cases, thus advancing understanding of nearly Kaehler geometry.

Contribution

It demonstrates that strict, complete nearly Kaehler manifolds decompose locally into products of known geometric spaces and classifies homogeneous cases, addressing a conjecture of Wolf and Gray.

Findings

Strict, complete nearly Kaehler manifolds are locally products of homogeneous spaces, twistor spaces, and 6-dimensional nearly Kaehler manifolds.

Structure results for Riemannian foliations with compatible Kaehler structures are established.

A classification for homogeneous nearly Kaehler manifolds reduces Wolf and Gray's conjecture to 6 dimensions.

Abstract

We consider strict and complete nearly Kaehler manifolds with the canonical Hermitian connection. The holonomy representation of the canonical Hermitian connection is studied. We show that a strict and complete nearly Kaehler is locally a Riemannian product of homogenous nearly Kaehler spaces, twistor spaces over quaternionic Kaehler manifolds and 6-dimensional nearly Kaehler manifolds. As an application we obtain structure results for totally geodesic Riemannian foliations admitting a compatible Kaehler structure. Finally, we obtain a classification result for the homogenous case, reducing a conjecture of Wolf and Gray to its 6-dimensional form.