Locally strongly convex affine hyperspheres realizing Chen's equality

Abstract

In affine differential geometry of hypersurface, C. Scharlach et al. found an inequality involving intrinsic and extrinsic curvatures, and classified elliptic and hyperbolic affine hyperspheres realizing the equality if an affine invariant 2-dimensional distribution D2 is integrable. In this paper, we continue to study affine hyperspheres realizing the equality, including parabolic affine hyperspheres. As main results, firstly we classify parabolic affine hyperspheres realizing the equality if its scalar curvature is constant, or D2 is integrable. Next, by introducing a well-defined 3-dimensional distribution D3 when D2 is not integrable, we complete the classification of locally strongly convex affine hyperspheres realizing the equality if D3 is integrable. Finally, we pose a conjecture and a problem in order to determine all affine hyperspheres attaining the equality.