Classification Of M?bius Parallel Submanifolds In The Unit Sphere

Geometry of submanifolds in the unit sphere is one of the most important research fields in differential geometry.Since professor Changping Wang has established the theory of M?bius geometry of submanifolds in 1998 [71],a lot of important progress and many interesting results were obtained in this field.Among them,we have the classification of submanifolds with particular M?bius invariants,but most of which are restricted to the hypersurface situation,and the classification of general codimension submanifolds are often much more complex and difficult.This paper mainly studies umbilic-free submanifolds of the unit sphere with parallel M?bius second fundamental form and the submanifolds with parallel M?bius form respectively,we also call the former are M?bius parallel submanifolds.Our main results can be stated as follows:Firstly,we give two kinds of typical examples and their M?bius characerizations,and then go into the study of M?bius parallel submanifolds,succeed in obtaining two crucial observations: the assumption “parallel M?bius second fundamental form”implies that the submanifold is of parallel Blaschke tensor,and the number t of distinct Blaschke eigenvalues is at most p + 2.The above results make it possible to classify the M?bius parallel submanifolds of general codimensions.Secondly,we completely classify the M?bius parallel submanifolds in the unit sphere.Make the best of the known results and our conclusions in chapter three,by means of algebraic techniques and moving frame method,at last,we finish the proof of classification theorem in each case,which generalize the result in [26] about hypersurfaces to submanifolds of arbitrary codimensions.Thus this problem is solved thoroughly.