The Study Of A Class Of Centroaffine Hypersurfaces

In the centroaffine geometry of hypersurfaces, there exist two important cen-troaffine invariants the difference tensor K and the centroaffine metric h which determine a hypersurface up to centroaffine transformations. In this paper, we study a class of centroaffine hypersurfaces which characterized by K(X1,X1)=?1X1, K(X1,X) = ?2X,K(X, Y) = ?2h(X, Y)X1, (?)X, Y (?) D2,where D1 (spanned by a unit vector field X1) and D2 are two mutually h-orthogonal differential distributions. Moreover, to study the class of centroaffine hypersurfaces,we need discuss two cases.The first case is A2 = 0, then Mn is an open part of a locally strongly convex hyperquadric with K 0 or (Mn,h) is locally isometric with a twisted product. Moreover, We achieve a complete classification of the class of centroaffine hypersurfaces which is locally isometric with a twisted product.The second one is ?2 ? 0, then (Mn,h) is locally isometric with a warped product. Moreover, we obtain a complete classification of such centroaffine hyper-surfaces.