On Weyl structures reducible in the direction of the Lee form

TL;DR

This paper investigates Weyl structures with a parallel distribution where the Lee form vanishes, proving flatness or exactness under completeness and closed Lee form conditions, and confirming a conjecture about homogeneous Kenmotsu manifolds.

Contribution

It establishes conditions under which Weyl structures are flat or exact and proves that all homogeneous Kenmotsu manifolds are isometric to real hyperbolic space.

Findings

Weyl structures with certain parallel distributions are flat or exact if complete and Lee form is closed.

Confirmed that every homogeneous Kenmotsu manifold is isometric to real hyperbolic space.

Provided new insights into the geometry of Weyl structures and Kenmotsu manifolds.

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.