Classification Of Calabi Hypersurfaces With Parallel Fubini-Pick Form

As we known,Fubini-Pick tensor plays an important role in the study of affine differential geometry.In this paper,we investigate the classification of low-dimensional Calabi hypersurfaces with parallel Fubini-Pick form with respect to the Levi-Civita connection of the Calabi metric,which is closely related to the classification of Tchebychev affine K（?）hler hypersurfaces and affine extremal hypersurfaces.In the first chapter,we provide the necessary background and relevant results.The basic facts and notions are presented in the second chapter.In the third chapter,we use integration to classify Calabi hypersurfaces with flat Calabi metric and parallel Fubini-Pick form with respect to the Levi-Civita connection of the Calabi metric.This theorem can be stated as follows:Theorem 3.1.Let f be a smooth strictly convex function on a domainΩ（?）Rn.If its graph M=｛（x,f（x））|x∈Ω｝has a flat Calabi metric and parallel Fubini-Pick form.Then M is Calabi affine equivalent to an open part of the following hypersurfaces:（1）elliptic paraboloid;（2）the hypersurfaces Q（c1,…,cr;n）,1 ≤r ≤n.The definitions of Calabi affine equivalent and hypersurfaces Q（c1,…,cr;n）will be given in the second chapter,see Definition 2.3 and Example 2.1,respectively.As a direct corollary of the above theorem,we get the classification of Calabi surfaces with parallel Fubini-Pick form with respect to the Levi-Civita connection of the Calabi metric.Corollary 3.1.Let f be a smooth strictly convex function on a domain Ω（?）R2.If its graph M=｛（x,f（x））| x∈Ω} has parallel Fubini-Pick form.Then M is Calabi affine equivalent to an open part of the following surfaces:（1）elliptic paraboloid;（2）the surfaces Q（c1,…,cr;2）,1≤r≤2.In the fourth chapter,motivated by the ideas in[23],we get the classification of the 3-dimensional case by solving the Gauss structure equations and with the help of de Rham decomposition theorem.Theorem 4.1.Let f be a smooth strictly convex function on a domain Ω（?）R3.If its graph M=｛（x,f（x））|x∈Ω} has parallel Fubini-Pick form.Then M is Calabi affine equivalent to an open part of one of the following three types of hypersurfaces:（1）elliptic paraboloid;（2）the hypersurfaces Q（c1,…,cr;3）,1≤r≤3;（3）the hypersurface x4=1/Rln（x12-（x22+x32））,where the constant R is the scalar curvature of M.In the fifth chapter,using a method similar to the theorem 4.1,we obtain the classification of the 4-dimensional case.Theorem 5.1.Let f be a smooth strictly convex function on a domain Ω（?）R4.If its graph M=｛（x,f（x））|x∈Ω}has parallel Fubini-Pick form.Then M is Calabi affine equivalent to an open part of one of the following five types of hypersurfaces:（1）elliptic paraboloid;（2）the hypersurfaces Q（c1,…,cr;4）,1≤r≤4;（3）the hypersurfaces（4）the hypersurfaces（5）the hypersurface x5=3/Rln（x12-（x22+x32+x42））,where the constant R is the scalar curvature of M,and the constant μ1 in（3）is related to the relative Pick invariant J and scalar curvature R byμ12=12J+7/2R.In the sixth chapter,we introduce the progress of related problems,and provide a sketch of the classification of high-dimensional Calabi hypersurfaces with parallel FubiniPick form.