Nearly-Kahler 6-Manifolds of Cohomogeneity Two: Local Theory

TL;DR

This paper classifies nearly-Kahler 6-manifolds with cohomogeneity-two group actions, establishing local existence results, analyzing metric completeness, and deriving PDE systems for certain types.

Contribution

It introduces a classification into three types, proves local existence for each, and analyzes the completeness and PDE conditions of these nearly-Kahler structures.

Findings

Types I and II are incomplete metrics.

Complete structures are quotients of S^3 x S^1.

Type III structures form a simple one-parameter family.

Abstract

We study nearly-Kahler 6-manifolds equipped with a cohomogeneity-two Lie group action for which the principal orbits are coisotropic. If the metric is complete, then we show that this last condition is automatically satisfied, and both the acting Lie group and the principal orbits are finite quotients of $S^3 \times S^1$. We partition the class of such nearly-Ka}hler structures into three types (called I, II, III) and prove a local existence and generality result for each type. Metrics of Types I and II are shown to be incomplete. We also derive a quasilinear elliptic PDE system on the 2-dimensional orbit space which nearly-Kahler structures of Type I must satisfy. Finally, we remark on a relatively simple one-parameter family of Type III structures that turn out to be incomplete metrics that are cohomogeneity-one under the action of a larger group.