Locally strongly convex affine hyperspheres realizing Chen's equality

[摘要] In afire 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 D-2 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 D-2 is integrable. Next, by introducing a well-defined 3-dimensional distribution D-3 when D-2 is not integrable, we complete the classification of locally strongly convex affine hyperspheres realizing the equality if D-3 is integrable. Finally, we pose a conjecture and a problem in order to determine all affine hyperspheres attaining the equality. (C) 2017 Elsevier Inc. All rights reserved.

[发布日期] 2017-12-15

[关键词] Affine differential geometry;Affine hypersphere;Chen's equality;delta(2)-invariant