Some Discussions On Riemannian Submanifolds With Parallel Ricci Curvature And Mobius Hypersurfaces In Sⁿ⁺¹

In this thesis,we discuss the rigid problems submanifolds inSn+pin theEuclidean space and the classification of hypersurfaces inSⁿ⁺¹in the Mobiusgeometry.This thesis consists of two chapters.In chapter1,we study the submanifolds with parallel mean curvature vector in aRiemannian manifolds with parallel Ricci curvature. We obtain an integral inequalityof simons’ type.The result of submanifolds with parallel mean curvature vector in alocally symmetric δ-pinching Riemannian manifold is generalized.In chapter2,we study the hypersurfaces with constant Mobius scalar curvatureand the para-Blaschke isoparametric hypersurfaces with three distinct para-Blaschkeeigenvalues inSⁿ⁺¹. Let:x:Mⁿâ†'Sⁿ⁺¹ï¼ˆn≥3）be a hypersurface in the（n+1）-dimensional unit sphereSⁿ⁺¹without umbilics. Four basic invariants of xunder the Mobius transformation group inSⁿ⁺¹are: a Riemannian metric g calledMobius metric,a1-form Φ called Mobius form, a symmetric （0,2）tensor A calledBlaschke tensor and a symmetric （0,2）tensor B called Mobius second fundamentalform. Let D^λ=A+λB, where λ is a constant. Then D is a symmetric （0,2）tensorand a Mobius invariant. D is called para-Blaschke tensor of x. In the first section,westudy the hypersurfaces with constant Mobius scalar curvature.We firstly define thetrace-free para-Blaschke tensor.We obtain an integral inequality of simons’ type bymaking use of it.Then we obtain a pinching theorem. In the second scetion,we give aclassification of the para-Blaschke isoparametric hypersurfaces with three distinctpara-Blaschke eigenvalues one of which is simple.