Pseudo-Parallel Characteristic Jacobi Operators on Contact Metric 3 Manifolds

Wenjie Wang,Xinxin Dai,Antonio Masiello

doi:10.1155/2021/6148940

Wenjie Wang, Xinxin Dai + Show 1 more

Open Access

https://doi.org/10.1155/2021/6148940

Copy DOI

Abstract

We prove that the characteristic Jacobi operator on a contact metric three manifold is semiparallel if and only if it vanishes. We determine Lie groups of dimension three admitting left invariant contact metric structures such that the characteristic Jacobi operators are pseudoparallel.

Highlights

Let (M2n+1, φ, ξ, η, g) be a contact metric manifold and l: R(·, ξ)ξ be the characteristic Jacobi operator associated with the characteristic vector field ξ, where R denotes the curvature tensor
The characteristic Jacobi operators were investigated by many authors and played important roles in the study of contact metric manifolds
Cho and Inoguchi in [9] classified all contact metric three manifolds such that ξ is an eigenvector field of the Ricci operator and the characteristic Jacobi operator is invariant along the Reeb flow, namely, Lξl 0, where L denotes the Lie differentiation

Summary

Introduction

Let (M2n+1, φ, ξ, η, g) be a contact metric manifold and l: R(·, ξ)ξ be the characteristic Jacobi operator associated with the characteristic (or Reeb) vector field ξ, where R denotes the curvature tensor. The characteristic Jacobi operators were investigated by many authors and played important roles in the study of contact metric manifolds. Koufogiorgos and Tsichlias in [8] classified all contact metric three manifolds with vanishing characteristic Jacobi operators. Cho and Inoguchi in [9] studied model spaces for contact metric three manifolds with vanishing characteristic Jacobi operators and constant |Qξ|. Cho and Inoguchi in [9] classified all contact metric three manifolds such that ξ is an eigenvector field of the Ricci operator and the characteristic Jacobi operator is invariant along the Reeb flow, namely, Lξl 0, where L denotes the Lie differentiation. We aim to investigate such problem and present that the characteristic Jacobi operators on contact metric three manifolds are semiparallel if and only if they are vanishing. We classify all left invariant contact metric structures on unimodular or nonunimodular Lie groups of dimension three such that the characteristic Jacobi operators are pseudoparallel. is shows that there exist no nontrivial semiparallel characteristic Jacobi operators, but there are nontrivial pseudoparallel characteristic Jacobi operators on contact metric three manifolds

Contact Metric Three Manifolds

Semiparallel Characteristic Jacobi Operators

Pseudoparallel Characteristic Jacobi Operators