Some Study On Critical Metrics Of Riemannian Curvature Functionals

Among the study of Riemannian geometry,variational problems for Rieman-nian functionals,manifolds' rigidity and solitons are always important subjects.In this paper,we focus on these problems and give some interesting results.We state our results in five chapters.In chapter one,we give some basic knowledge and important theories of Rie-mannian geometry which are necessary for our paper.Lately we state the back-ground of our research and the main results we obtain.In chapter two,we focus on a the Bach-flat critical metrics of Riemannian functionalHere FT are defined on close manifolds and constrainted on unit volume metrics space M1.Under different dimension,we find some conditions for those critical metrics become Einstein.In chapter three,we consider quadratic curvature functionalwhich is defined on closed Riemannian manifolds with unit volume metric.We give a rigidity result for complete noncompact Riemannian manifolds with cyclic parallel Ricci tensor and some properties of Miao-Tam critical metric with with cyclic parallel Ricci tensor in chapter four.In chapter five,we concentrate on the h-almost Yamabe soliton which is gener-alization of classical Yamabe soliton.Two conditions are given for the vector field among h-almost Yamabe soliton to be Killing.Meanwile,we study the classification of h-almost Yamabe soliton on hypersurfaces in Euclidean space.