On Weyl structures reducible in the direction of the Lee form

Abstract

A Weyl structure on a Riemannian manifold $(M,g)$ is a torsion-free linear connection $\nabla$ such that there is a $1$-form $\theta$ (called the Lee form) satisfying $\nabla g = 2\, \theta \otimes g$. We examine the case in which there exists a $\nabla$-parallel distribution of codimension $1$ on which the Lee form vanishes identically. We prove that if $(M,g)$ is complete with $\theta$ closed, then the Weyl structure must be flat or exact. We apply this to prove the conjecture of Lotta (Eur. J. Math., 2023), namely, every homogeneous Kenmotsu manifold is isometric to the real hyperbolic space.