Submanifolds In Real Space Forms With Parallel Blaschke Tensor

This dissertation discusses the conformal differential geometry of submanifolds in the real space forms,with the emphasis on the complete classifications for hypersurfaces or submanifolds which have some parallel conformal invariants.It consists of two main parts, one of which is the classification of regular space-like hypersurfaces in the de Sitter space with parallel para-Blaschke tensor,the other being the classification of general umbilic-free submanifolds in the anit sphere with parallel Blaschke tensor and vanishing Mobius form.The main theorems are obtained as follows:Theorem 0.1?See Theorem 1.3 in chapter One?Let x:M^m?S₁^m+1,m?2,be a regular space-like hypersurface in the de Sitter space S₁^m+1. Suppose that,for some constant?,the corresponding para-Blaschke tensor D^?:=A+?B of x is parallel.Then x is locally conformal equivalent to one of the following hypersurfaces:1.a regular space-like hypersurface in S₁^m+1.with constant scalar curvature and constant mean curvature;2.the image under ?o ?₀ of a regular space-like hypersurface in R₁^m+1 with constant scalar curvature and constant mean curvature;3.the image under ? o ?_-1 of a regular space-like hypersurface in H₁^m+1 with constant scalar curvature and constant mean curvature;4.s^m-k?a?×H^k?-1/?a²-1?????S₁^m+1,a>1,k=1,...m-1;5.the image under ? o ?₀ of H^k?-1/a2?×Rm-k???R₁^m+1,a>0,k=1,…m-1;6.the image under ? o ?-1 of Hk?-1/a2?×H^m-k?-1/?a²-1?????H₁^m+1,0<a<1,k= 1,…m-1;7. the image under ? of W P?p, q, a? ??? Q₁^m+1 for some constants p, q, a ?see Example 3.1?;8. one of the regular space-like hypersurfaces as indicated in Example 3.2;9. one of the regular space-like hypersurfaces as indicated in Example 3.3.Theorem 0.2 ?See Theorem 1.9 in Chapter One? Let x:M^m?S^m+p be a Blaschke parallel submanifold immersed in S^m+p. If x is of vanishing Mobius form C, then it must be Mobius equivalent to one of the following four kinds of immersed submanifolds:?1? a pseudo-parallel and umbilic-free immersion x:M^m ? S^m+p with parallel mean curvature and constant scalar curvature;?2? the image under ? of a pseudo-parallel and umbilic-free immersion x:M^m ? R^m+p with parallel mean curvature and constant scalar curvature;?3? the image under ? of a pseudo-parallel and umbilic-free immersion x:M^m ? H^m+p with parallel mean curvature and constant scalar curvature;?4? a submanifold LS?m₁,p₁,r,?? given in Example 4.2 for some certain parameters m₁, p₁, ?,? or a submanifold LS?m, p,?,?? given in Example 5.2 for a certain set of multiple parameters m. p, ?,?.The dissertation is divided into five chapters. The organizing structure is as follows:Chapter One is an introduction consisting of two sections and presents the back-ground and the main results.Chapter Two is the preliminary material divided into two sections:In Section One we introduce the conformal differential geometry of submanifolds in the Lorentz space forms, including the fundamental conformal invariants; While in Section Two we give the Mobius geometry of submanifolds in the unit sphere, emphasizing the fundamental Mobius invariants.Chapter Three aims at the classification of regular space-like hypersurfaces in the de Sitter space with parallel para-Blaschke tensor. This chapter has two sections:In Section One we give new examples satisfying all the desired conditions, and in Section Two we prove Theorem 0.1.Chapter Four and Chapter Five devote to the classification of the general umbilic-free submanifolds in the unit sphere with parallel Blaschke tensor and vanishing Mobius form. Concretely, in Chapter Four we consider classification theorems for two special cases, namely, submanifolds with respectively two and three distinct Blaschke eigenvalues. There are three sections here:The first introduces new examples, while the second and the third aim at proving the classification theorems respectively for the two special cases ?see Theorem 1.6 and Theorem 1.7?; While in Chapter Five we first give a class of more general examples ?in Section One? and then, in Section Two we prove the classification theorem ?Theorem 1.8? for submanifolds with t?? 4? distinct Blaschke eigenvalues. Finally, on the basis of known conclusions we complete the proof of the main theorem ?Theorem 0.2?.