Para-Ricci-like Solitons with Vertical Potential on Para-Sasaki-like Riemannian Π-Manifolds

Hristo Manev

doi:10.3390/sym13122267

Abstract

The objects of study are para-Ricci-like solitons on para-Sasaki-like, almost paracontact, almost paracomplex Riemannian manifolds, namely, Riemannian Π-manifolds. Different cases when the potential of the soliton is the Reeb vector field or pointwise collinear to it are considered. Some additional geometric properties of the constructed objects are proven. Results for a parallel symmetric second-order covariant tensor on the considered manifolds are obtained. An explicit example of dimension 5 in support of the given assertions is provided.

Highlights

We study the covariant derivative of the Ricci tensor with respect to the metric g of a (2n + 1)-dimensional para-Sasaki-like Riemannian Π-manifold (M, φ, ξ, η, g) with a para-Ricci-like soliton of the considered type
Let (M, φ, ξ, η, g) be a para-Sasaki-like Riemannian Π-manifold of dimension 2n + 1 and let it admit a para-Ricci-like soliton with constants (λ, μ, ν) whose potential vector field v satisfies the condition v = k ξ, i.e., it is pointwise collinear with the Reeb vector field ξ, where k is a differentiable function on M
The objects of study were para-Ricci-like solitons on para-Sasaki-like Riemannian Πmanifolds

Summary

Introduction

The investigation of the Ricci solitons in different types of almost contact metric manifolds were done in [4,5,6]. Different generalizations of this concept were studied: in paracontact geometry [7,8], in pseudo-Riemannian geometry [9,10,11,12,13,14].

Para-Sasaki-like Riemannian Π-Manifolds

Some Additional Curvature Properties

Example

Conclusions