Some Results on Ricci Almost Solitons

Sharief Deshmukh,Hana Alsodais,Nasser Bin Turki

doi:10.3390/sym13030430

Abstract

We find three necessary and sufficient conditions for an n-dimensional compact Ricci almost soliton (M,g,w,σ) to be a trivial Ricci soliton under the assumption that the soliton vector field w is a geodesic vector field (a vector field with integral curves geodesics). The first result uses condition r2≤nσr on a nonzero scalar curvature r; the second result uses the condition that the soliton vector field w is an eigen vector of the Ricci operator with constant eigenvalue λ satisfying n2λ2≥r2; the third result uses a suitable lower bound on the Ricci curvature S(w,w). Finally, we show that an n-dimensional connected Ricci almost soliton (M,g,w,σ) with soliton vector field w is a geodesic vector field with a trivial Ricci soliton, if and only if, nσ−r is a constant along integral curves of w and the Ricci curvature S(w,w) has a suitable lower bound.

Highlights

Given a Riemannian manifold (M, g), the sectional curvature distributions divide the manifold into three portions, one where sectional curvatures are positive, another where sectional curvatures are negative and the third, where sectional curvatures are zero.Hamilton, in his quest to solve Poincare conjecture, realized the role of a heat equation that evenly distributes temperature on the region, and considered a heat equation for the evolving metric known as Ricci flow, for an excellent description on this topic, we refer to
In [3], the authors considered the stable solution of the Ricci flow of the form gt = f (t, x)ψt∗(g)
We show that for a compact (M, g, w, σ) with w a geodesic vector field and nonzero scalar curvature r satisfying r2 ≤ nσr is necessary and sufficient to be trivial

Summary

Introduction

Given a Riemannian manifold (M, g), the sectional curvature distributions divide the manifold into three portions, one where sectional curvatures are positive, another where sectional curvatures are negative and the third, where sectional curvatures are zero. A Ricci almost soliton could be an Einstein manifold without being trivial, as suggested by the example (Sn(c), g, w, σ), where Sn(c) is the sphere of constant curvature c and w = grad h for some smooth function on the sphere (cf [4,11]). We show that if a compact (M, g, w, σ) with w a geodesic vector field satisfies Q(w) = λw for a constant λ with r2 ≤ n2λ2, if and only if, it is a trivial Ricci soliton (cf Theorem 2). We show that for a connected (M, g, w, σ) with w a geodesic vector field and Ricci curvature S(w, w) has certain lower bound and the function nσ − r is a constant on integral curves of w if and only if (M, g, w, σ) is a trivial Ricci soliton (cf Theorem 4)

Preliminaries

Ricci Almost Solitons with Geodesic Soliton Vector Field

Conclusions