DOI: 10.36890/iejg.1737453 ISSN: 1307-5624

On Locally Conformal Spin(7) Structures

Eyüp Yalçınkaya
Show PDF Cite
In this article, we investigate the geometric interplay between locally conformal $\mathrm{Spin}(7)$ structures on 8-dimensional manifolds and induced $\mathrm{SU}(3)$-structures on 6-dimensional submanifolds. We show that if the Lee form $\theta$ of a locally conformal $\mathrm{Spin}(7)$ manifold is normal to a 6-dimensional oriented submanifold $N$, then the induced $\mathrm{SU}(3)$-structure on $N$ belongs to the Gray–Hervella class $\mathcal{W}_4$, i.e., it is locally conformal almost Hermitian. Conversely, we prove that if the induced $\mathrm{SU}(3)$-structure lies purely in $\mathcal{W}_4$, then the Lee form must be normal to $N$. This establishes a precise correspondence between the direction of the Lee form and the intrinsic torsion of the induced $\mathrm{SU}(3)$-structure. Furthermore, we construct explicit examples of locally conformal $\mathrm{Spin}(7)$ manifolds with two commuting vector fields that realize this correspondence. Our results provide a systematic framework for deriving 6-dimensional geometric structures from higher-dimensional ambient geometries and contribute to the broader program of understanding torsion phenomena in manifolds with special holonomy.

More from our Archive