Some Rigidity Theorems For Riemannian Manifolds With Special Geometric Structures

It is a very important subject to study classification and rigidity theorems on Rieman-nian manifolds with special geometric structures in differential geometry.In this disser-tation,we mainly consider classifications and rigidity characterisations with the following geometric structures:generalized Bach-flat four dimensional manifolds;Bach-flat mani-folds of dimension n≥4;manifolds with harmonic Riemannian curvature tensor;rigidity characterisations on critical metrics of quadratic curvature functionals.We always study the following two kinds of rigidity characterisations in each part of the paper:compact Riemannian manifolds and complete Riemannian manifolds,respectively.The main meth-ods used are the following:By analysing and comparing special geometric structures,we obtain some sharper estimates(for example,see(2-30)in Lemma 2.2.9);Using the maxi-mum principle(see(1-11)in Lemma 1.3.7)etc.,we deduce the desired results from some well-known classical theoriesThis dissertation is organized as followsIn Chapter One,some related backgrounds,necessary preliminaries and several lem-mas are introducedIn Chapter Two,Chapter Three and Chapter Four,we study,respectively,generalized Bach-flat manifolds of dimension four,Bach-flat manifolds of dimension n≥4;manifolds with harmonic Riemannian curvature tensor.Using inequalities with respect to the Weyl curvature,the Ricci curvature and the scalar curvature,we obtain some rigidity char-acterisations for Einstein and,in particular,for those manifolds with constant sectional curvature.Moreover,by virtue of the integral inequalities involving the Weyl curvature,the Ricci curvature and Yamabe invariant,we also obtain some similar results.Especially,some of the above results generalize a number of known ones.In Chapter Five,we study rigidity characterisations of the critical metrics of the following quadratic curvature functionals:Ft=(?)|Rij|2+t(?)R2,t∈R.M MAs the result,we obtain some rigidity characterisations of Einstein manifolds and mani-folds with constant sectional curvature,by using point-wise estimates with respect to the Weyl curvature,the Ricci curvature and the scalar curvature.On the other hand,with the help of integral formulas of compact manifolds and critical metrics,we give some corre-sponding rigidity characterisations.In particular,for complete manifolds,we obtain some rigidity results by using point-wise inequalities with respect to the Weyl curvature,the Ricci curvature and the scalar curvature,which generalize those corresponding results in Huang-Chen-Li[59].