QR-submanifolds and Riemannian metrics with $G_2$ holonomy

TL;DR

This paper demonstrates that certain QR-submanifolds of hyper-Kahler manifolds can admit $G_2$ holonomy, providing explicit examples like tori and conjecturing a broader classification of all such manifolds.

Contribution

It establishes conditions under which QR-submanifolds of hyper-Kahler manifolds have $G_2$ holonomy and introduces the conjecture that all $G_2$ holonomy manifolds can be obtained this way.

Findings

QR-submanifolds of hyper-Kahler manifolds can admit $G_2$ holonomy

Explicit examples include tori

Conjecture that all $G_2$ holonomy manifolds arise as such QR-submanifolds

Abstract

In this note we prove that QR-submanifolds of the hyper-Kahler manifolds under some conditions admit the $G_2$ holonomy. We give simplest examples of such QR-submanifolds namely tori. We conjecture that all $G_2$ holonomy manifolds arise in this way.