The Calabi Decomposition Of The Centroaffine Hypersurfaces

In the centroaffine differential geometry,there exist two important centroaffine invariants the centroaffine metric h and the difference tensor K.In this paper,we study the Calabi decomposition problem of the locally strongly convex centroaffine hypersurfaces which are characterized by:K_TT=?₁T,K_TX=?₂X,K_TY=?₃Y,K_XY=0,T?D₁,?X?D₂,?Y?D₃,where D₁?spanned by a unit vector field T?,D₂?spanned by the vector field X?and D₃?spanned by the vector field Y?are three mutually orthogonal differential distributions.In order to study this problem,there are four steps in this paper as follows:Step 1:Using the the integrability condition?2.4?to get a series of equations?See lemma 3.2 to lemma 3.6?about vector field T,X,Y for the centroaffine metric h.Step 2:In order to ensure that the three distributions D₁,D₂and D₃are inte-grable,we add the two conditions grad?₁?D₁and h?[X,Y],T?=0?See lemma 3.7?.At the same time,it is easy to see that D₁?D₂and D₁?D₃are autoparallel.Step 3:Having solved the integrability problem about the three distributions,we can use the results got by the first two steps to get that grad?₂and grad?₃are both parallel to T.Then we can solve the problem that D₂and D₃are spherical.From the decomposition theorem 2.4 of Riemannian manifolds and the results got by the first three steps,it is enough to see that the locally strongly convex centroaffine hypersurfaces and the warped product R×_?2M₂×_?3M₃are isometric.Step 4:Adding the two conditions K_D2D₂?D₂,K_D3D₃?D₃,using the results got by the first three steps and the integrability condition?2.3?,comput-ing the first and the second covariant derivative about functions,we can prove that the locally strongly convex centroaffine hypersurfaces?Mⁿ,h?can be Calabi decomposed into two proper affine hyperspheres?See the proof of the main result?.