Locally conformal parallel $G_2$ and $Spin(7)$ manifolds

TL;DR

This paper characterizes compact locally conformal parallel $G_2$ and $Spin(7)$ manifolds as fiber bundles over the circle with specific nearly Kähler or nearly parallel $G_2$ fibers, especially when structures are flat.

Contribution

It provides a new geometric characterization of these special manifolds as fiber bundles over $S^1$ with particular fiber structures.

Findings

Manifolds are fiber bundles over $S^1$ with nearly Kähler or nearly parallel $G_2$ fibers.

A detailed description when the local structures are flat.

Enhanced understanding of the geometric structure of conformal parallel manifolds.

Abstract

We characterize compact locally conformal parallel $G_2$ (respectively, $Spin(7)$) manifolds as fiber bundles over $S^1$ with compact nearly K\"ahler (respectively, compact nearly parallel $G_2$) fiber. A more specific characterization is provided when the local parallel structures are flat.