On Locally Conformal Spin(7) Structures

In this article, we investigate the geometric interplay between locally conformal Spin(7) structures on 8-dimensional manifolds and induced SU(3)-structures on 6-dimensional submanifolds. We show that if the Lee form θ of a locally conformal Spin(7) manifold is normal to a 6-dimensional oriented submanifold N, then the induced SU(3)-structure on N belongs to the Gray–Hervella class W₄, i.e., it is locally conformal almost Hermitian. Conversely, we prove that if the induced SU(3)-structure lies purely in W₄, 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 SU(3)-structure. Furthermore, we construct explicit examples of locally conformal 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.