A Note on the Characterization of Two-Dimensional Quasi-Einstein Manifolds

Gabriel Bercu

doi:10.3390/math8112013

Abstract

In this article, we aim to introduce new classes of two-dimensional quasi-Einstein pseudo-Riemannian manifolds with constant curvature. We also give a classification of 2D quasi-Einstein manifolds of warped product type working in local coordinates. All the results are obtained by elementary methods.

Highlights

Nonlinear differential equations arise in wide areas of research in pure and applied sciences.In this note, we study some non-linear differential equations whose solutions are quasi-Einstein metrics that appear in mathematical physics [1,2]
We study some non-linear differential equations whose solutions are quasi-Einstein metrics that appear in mathematical physics [1,2]
The quasi-Einstein metric is a generalization of the Einstein metric, it contains gradient Ricci solitons and it is closely related to the construction of the warped product Einstein metrics

Summary

Introduction

Nonlinear differential equations arise in wide areas of research in pure and applied sciences. Given a smooth real valued mapping F, defined on M and μ a real constant, suppose that the tensor having the components hij = F,ij − μF,i F,j is non-degenerate and has constant signature With these assumptions, we establish a link between the Christoffel symbols Γijk of k h and the Christoffel symbols Γij of the initial metric g. The relation which defines the quasi-Einstein pseudo-Riemannian metrics: Rij + F,ij − μF,i F,j = ρgij can be written as: F,ij − μF,i F,j = [ρ − (n − 1) K ] gij This means that the Christoffel symbols Γijk of h and the Christoffel symbols Γij of g have to be identical. We consider the warped product Riemannian metrics g( x, y) = diag( g(y), 1), with constant sectional curvature, and we obtain a complete classification of those metrics which are quasi-Einstein too

A New Class of Quasi-Einstein Metrics

Generalizing a Question of Besse

Conclusions

Methods