Classification Of Gradient Almost Ricci Solitons Under The Pinching Conditions

By using the method of geometric analysis,divergence theorem and some important inequalities,in this paper,we study the classification of gradient almost Ricci solitons under different conditions,and obtain local characterization and rigid results of gradient almost Ricci solitons.The main results obtained in this paper include the following three parts.1.we investigate the local characterization of four-dimensional complete gradient almost Ricci solitons under the certain pointwise pinching conditions involving either the anti-self-dual or self-dual part of the Weyl tensor,and prove that such solitons are locally warped product with three-dimensional fibers of constant sectional curvature or Einstein fibers.2.Under the assumption that the scalar curvature is positive,we prove that n-dimensional,n≥ 4,compact gradient almost Ricci solitons satisfying certain Ln/2-pinching conditions are Einstein manifolds.In particular,under the analogous pinching condition,we prove that the four-dimensional compact gradient almost Ricci solitons is isometric to a finite quotient of the round sphere S4.3.Similar to the ideas and methods of the second part,we prove that ndimensional,n≥ 4,complete gradient almost Ricci solitons are isometric to finite quotient of Rn or Sn under some appropriate conditions.