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On Weyl structures reducible in the direction of the Lee form

José Luis Carmona Jiménez

Abstract

A Weyl structure on a Riemannian manifold $(M,g)$ is a torsion-free linear connection $\nabla$ such that there is a $1$-form $θ$ (called the Lee form) satisfying $\nabla g = 2\, θ\otimes g$. 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 $θ$ 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.

On Weyl structures reducible in the direction of the Lee form

Abstract

A Weyl structure on a Riemannian manifold is a torsion-free linear connection such that there is a -form (called the Lee form) satisfying . We examine the case in which there exists a -parallel distribution of codimension on which the Lee form vanishes identically. We prove that if is complete with 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.

Paper Structure

This paper contains 7 sections, 12 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definition and examples
  4. Main Result
  5. The compact case
  6. Applications
  7. Homogeneous Kenmotsu Manifolds

Key Result

theorem 1.1

Let $(M,g)$ be a complete Riemannian manifold, and let $\nabla$ be a closed Weyl structure reducible in the direction of the Lee form. Then $\nabla$ is flat or exact.

Theorems & Definitions (25)

  • theorem 1.1
  • conjecture 1.2
  • definition 3.1
  • example 3.2
  • example 3.3
  • definition 3.4
  • remark 3.5
  • lemma 4.1
  • proof
  • theorem 4.2: PG1993
  • ...and 15 more