On Non-degenerate Affine Hypersurfaces With Parallel Cubic Form

In this paper, we study non-degenerate affine hypersurfaces with parallel cubic form. Our main results are as follows:(1) We characterize the Calabi composition of affine hyperspheres under the condi-tion of parallel cubic form, which improves the results of Z.J. Hu, C.C. Li, H.Z. Li and L. Vrancken whose results were obtained for locally strongly convex or Lorentzian affine hypersurfaces;(2) For given integers m,n with m+1=n(n+1)/2, we establish the standard hypersurface embedding from SL(n, R)/SO(p, q;R) into the affine space Rm+1. Moreover, we study its Blaschke structure and show that it is an Einstein affine hypersurface which is of signature (m-pq, pq) and having parallel cubic form:▽C=0;(3) Giving the existence of a non-zero null vector V with K (V, V)=0, we partially classify the non-degenerate affine hypersurfaces with parallel cubic form (with respect to the Levi-Civita connection) in R6, showing that one of them is locally affmely equivalent to the standard embedding SL(3,R)/SO(1,2;R)→R6.