Classification of Warped Product Submanifolds in Kenmotsu Space Forms Admitting Gradient Ricci Solitons

Ali Alkhaldi,Akram Ali

doi:10.3390/math7020112

Abstract

The purpose of this article is to obtain geometric conditions in terms of gradient Ricci curvature, both necessary and sufficient, for a warped product semi-slant in a Kenmotsu space form, to be either CR-warped product or simply a Riemannian product manifold when a basic inequality become equality. The next purpose of this paper to find the necessary condition admitting gradient Ricci soliton, that the warped product semi-slant submanifold of Kenmotsu space form, is an Einstein warped product. We also discuss some obstructions to these constructions in more detail.

Highlights

Introduction and Motivations of the MainResultsAs a generalization of Riemannian product manifolds, the warped product manifolds are defined as follows: Definition 1
We consider the following question: “What are necessary and sufficient conditions for warped product immersions in Kenmotsu space forms to be an Einstein warped product manifold with the impact of gradient Ricci soliton by using inequality (1)?” The answer, by assuming the equality case in the inequality (1) and a vector field X is the gradient of the warping function of warped product manifold
We will discuss some geometric applications in various physical terms such as Euler-Lagrange equation, Ricci curvature and divergence of Hessian

Summary

Introduction and Motivations of the Main Results

As a generalization of Riemannian product manifolds, the warped product manifolds are defined as follows: Definition 1. We consider the following question: “What are necessary and sufficient conditions for warped product immersions in Kenmotsu space forms to be an Einstein warped product manifold with the impact of gradient Ricci soliton by using inequality (1)?” The answer, by assuming the equality case in the inequality (1) and a vector field X is the gradient of the warping function of warped product manifold. Assume that χ : Mn = NT1 × f Nθn is an isometric immersion of a compact warped product n e 2m+1 (c) with satisfying the following equality semi-slant submanifold NT1 × f Nθn into Kenmotsu space form M for warped product submanifold Mn If the warping function has solution of the Euler-Lagrange semi-slant into Kenmotsu space form M n equation, the necessary and sufficient condition of the warped product NT1 × f Nθn is trivial such that. We shall prove other results related to the above study

Preliminaries

Main Results for Warped Product Semi-Slant and Their Applications

Proof of the Theorem 5

Proof of the Theorem 3

Proof of the Theorem 4