Lee form and special warped-like product manifolds with locally conformally parallel Spin(7) structure

Abstract

We study the cases of the Lee form on special warped-like product manifolds M with locally conformally parallel Spin(7) structure to determine the nature of the fibers. Using fiber-base decomposition, we prove that the connection on M is determined by the Bonan form and Lee one-form. Assuming that the fibers are complete, connected and simply connected, and choosing some classes of Lee form on M, we prove a main result that the fibers (or at least one of them) are isometric to S3 with constant curvature k > 0 in the class of (3 + 3 + 2) warped-like product metrics admitting a specific locally conformally parallel Spin(7) structure. We believe that the paper could help in producing new examples of (locally conformally) parallel Spin(7) structures.