Equiaffine Isopermetric Functions And Their Regular Level Hypersurfaces

In this paper,we introduce and study the locally strongly convex equiaffine isopara-metric hypersurfaces and equiaffine isoparametric functions on the affine space An+1 The main content is divided into two parts.In the first part,we first introduce the concept of equiaffine parallel hypersurfaces in An+1,obtaining some fundamental identities with the basic equiaffine geometric invariants,and then we define the equiaffine isoparametric hypersurfaces to be ones that are among families of equiaffine parallel hypersurfaces of constant affine mean curvature in An+1.In the second part,we introduce the concept of equiaffine isoparametric functions on An+1,and prove that any equiaffine isopaaametric hypersurface is exactly a regular level set of some equiaffine isoparametric function.This dissertation is divided into four chapters:Chapter One is an introduction presenting the background and the main results.Chapter Two is the preliminary material,which is divided into two sections:In Section One we introduce the definitions and notations which we will use later;While in Section Two we brief some basic facts for the equiaffine differential geometry of locally strongly convex hypersurfaces.In Chapter Three we introduce the concept of equiaffine parallel hypersurfaces in the affine space An+1(Definition 3.2),prove other equivalent definitions(Proposition 3.1 and Corollary 3.2),and obtain some fundamental identities.Moreover,we introduce the concept of equiaffine isoparametric hypersurface(in Definition 3.3).In Chapter Four we first introduce the concept of equiaffine isoparametric functions on An+1(Definition 3.3),and then prove the main theorems.