Studies On The Rigidity Of Almost Ricci Solitons

In this thesis,we mainly discuss the rigidity of almost Ricci solitons,by using the Obata equation,divergence theorem,cutoff function and other methods.Under the assumption that the soliton vector field is a conformal vector field,or the fourth-order divergence of the Weyl tensor is nonnegative,or a pointwise pinching condition,we obtain some rigid results of almost Ricci solitons.The specific contents are as following:1.We study the proper almost Ricci solitons with the conformal soliton vector field,by using the method of Obata equation,and deduce that such a soliton either is isometric to one of Rn,Sn,and Hn or isometric to a warped manifold R×?M with(n-1)-dimensional Einstein fiber M.2.We investigate the rigidity of gradient almost Ricci soliton under the the premise of having nonnegative fourth order divergence on Weyl tensor.More pre-cisely,we prove such a gradient almost Ricci soliton is locally a warped product manifold with Einstein fibers.3.Using again the ideas and methods in the second part,we study the rigid-ity of gradient(?,m)-Quasi-Einstein solitons under the the premise of admitting nonnegative fourth order f-divergence on Weyl tensor.We prove that such a gra-dient(?,m)-Quasi-Einstein solition has harmonic Weyl tensor.Furthermore,under a geometric assumption,we prove that a compact gradient m-Quasi-Einstein mani-fold with nonnegative fourth order f-divergence on Weyl tensor,is locally a warped product manifold.4.By calculating the X-Laplacian for the squared norm of the trace-free curva-ture tensor,we study the rigidity problem of almost Ricci solitons under a pointwise pinching condition.On the assumption that the scalar curvature is nonnegative,we prove that a complete almost Ricci soliton is isometric to a finite quotient of Rn or Sn under a pointwise pinching condition.For a compact almost Ricci soliton,we obtain an integral inequality and prove that the equality occurs if and only if the soliton is isometric to a finite quotient of Sn.