Triviality and Rigidity of Almost Riemann Solitons

Abstract

In this paper, we study some triviality and rigidity results of Riemann soliton. First, we derive some sufficient conditions for which an almost Riemann soliton is trivial. In particular, we prove that any compact almost Riemann soliton with constant scalar curvature has constant sectional curvature. Next, we prove some rigidity results for gradient Riemann solitons. Precisely, we prove that a non-trivial gradient Riemann soliton is locally isometric to a warped product \(( I \times F, \textrm{d}t^2 + f(t)^2g_{F})\), where \(\nabla \sigma \ne 0\).