On The Rigidity And Classification Problems Of Affine Hypersurfaces

As a topic of great significance in affine differential geometry,the research on the rigidity and classification problems of affine hypersurfaces has been paid much attention by geometers.In this dissertation,equiaffine hypersurfaces and centroaffine hypersurfaces are respectively used to carry out the above research,and the main results are as follows:（1）We derive a rigidity theorem for equiaffine hyperovaloids in affine space Rⁿ⁺¹.Firstly,we shall calculate the Laplacian of the Pick invariant on equiaffine hypersurfaces and arrange it into a special form as needed.Secondly,making use of the techniques of construction and combination,we give the lower bound esti-mates of some related equiaffine invariants.Finally,on equiaffine hyperovaloids,we establish an optimal integral inequality between the equiaffine intrinsic and extrinsic invariants,and prove that its equality holds if and only if these hyper-ovaloids are ellipsoids.In addition,through studying the locally strongly convex equiaffine hypersurfaces with semi-parallel cubic form,we obtain new local and global characterizations of the ellipsoids.（2）We complete the classification of 4-dimensional Lorentzian affine hyper-spheres with Einstein affine metric and non-zero Pick invariant.The unit tangent bundle of the above affine hyperspheres is no longer compact at a point,which is essentially different from the proof in[71]for locally strongly convex case.To overcome this difficulty,we utilize the compactness of the set,consisting of all null-directions in the light-cone of the tangent space,to construct the typical null frames in different cases.By proving that the above affine hyperspheres must be of constant sectional curvature and basing on the known conclusions,we completely determine such affine hyperspheres.（3）We derive a rigidity theorem for centroaffine hyperovaloids in affine space Rⁿ⁺¹.Considering the centroaffine invariants,we apply the classical Yano formula in Riemannian geometry to obtain an optimal integral inequality on such hyper-ovaloids.By calculating the corresponding invariants of the ellipsoids,we prove that the equality in the above inequality holds if and only if these hyperoval-oids are the ellipsoids such that the origin of Rⁿ⁺¹is located in their interior.Moreover,we construct isoparametric functions on such ellipsoids.（4）We complete the classification of locally strongly convex centroaffine hy-persurfaces with constant sectional curvature and vanishing centroaffine shape operator.According to the typical properties of Riemannian manifolds with closed conformal vector fields,we completely determine such hypersurfaces by further calculating the Laplacian of the Pick invariant with respect to the cen-troaffine metric.Furthermore,applying this classification and the calculation of canonical locally strongly convex centroaffine hypersurfaces,we completely clas-sify 4-dimensional flat hyperbolic centroaffine Tchebychev hypersurfaces.