Conformally flat centroaffine Tchebychev hypersurfaces

In this paper, we study locally strongly convex, conformally flat centroaffine hypersurfaces. The key step is to obtain some relations between the Schouten operator P, the difference tensor and its traceless tensor, which is achieved by applying the Tsinghua principle for two related symmetric Codazzi tensors of (1, 2)-form on a conformally flat manifold. As the main results, (1) such hypersurfaces are proved to be of constant sectional curvature if P has no eigenvalues of multiplicity one; (2) such Tchebychev hypersurfaces are classified if P has exactly one eigenvalue of multiplicity one. As a byproduct, we obtain the classification of conformally flat proper affine hyperspheres whose Schouten operator admits at most one eigenvalue of multiplicity one.