Rigidity Results Of H-almost Ricci Solitons

In this thesis,we mainly investigate h-almost Ricci solitons and gradient h-almost Ricci solitons on Riemannian manifolds and special Riemannian manifold-Einstein manifolds.By using the some identity,divergence theorem,Schur lemma and Hopf's strong extremum principle on Riemannian manifolds,the rigidity of h-almost Ricci solitons and gradient h-almost Ricci solitons are discussed.The specific conclusions are as following.1.Compact h-almost Ricci solitons and gradient h-almost Ricci solitons are studied.Firstly,we prove that the compact h-almost Ricci solitons with a nontriv-ial conformal vector field isometric to a Euclidean sphere Sm.Secondly,by using the basic equation of Riemannian manifold and the second Bianchi identity of con-traction,we prove that the compact gradient h-almost Ricci solitons which satisfy the proper integral conditions isometric to a Euclidean spheres Sm.Finally,By us-ing the divergence theorem and Schur lemma,we prove that the compact h-almost Ricci solitons and the gradient h-almost Ricci solitons satisfying the proper integral conditions are trivial.2.The complete noncompact h-almost Ricci solitons and gradient h-almost Ricci solitons are studied.Using the research method of the conclusion in the first part and the Hopf's strong extremum principle,we obtain that the complete noncompact h-almost Ricci soliton whose Ricci curvature satisfies the appropriate inequality conditions is trivial.Also,the complete noncompact gradient h-almost Ricci soliton satisfying the appropriate inequality conditions is trivial.3.The complete gradient h-almost Ricci solitons on special Riemannian manifold-Einstein manifold is studied.By using Ricci identity and suppose that the Ricci curvature on Einstein manifold vanishes identically,we prove that the complete gradient h-almost Ricci soliton is isometric to a Rm.