Locally strongly convex affine hypersurfaces that are generalized Calabi products

Abstract

In affine differential geometry, there is a well known construction, discovered by Calabi, of obtaining a new affine hyperbolic hypersphere starting with two hyperbolic affine hyperspheres or a hypersphere and a point. Intrinsically, the new hypersphere is the product of two original hyperspheres and a one-dimensional factor with image which is a special planar curve. This construction can be generalized by taking arbitrary planar curve. Such hypersurfaces admit a special decomposition of their tangent bundle into two or three distributions, defined in terms of the shape operator and the difference tensor and have a particular warped product structure. We will present the characterisation of the locally strongly convex affine hypersurfaces that can be decomposed as a generalized Calabi product of two affine hyperspheres or a hypersphere and a point. Further, we will study hypersurfaces with the constant sectional curvature whose shape operator has at most one eigenvalue of multiplicity one, show that they admit the generalized Calabi decomposition and give their classification. The results are obtained in collaboration with F. Dillen, Z. Hu, C. Li, H. Li, K. Schoels, L. Vrancken and X. Wang.