Abstract
In this paper we prove that an affine hypersphere with scalar curvature zero in a unimodular affine space of dimensionn+1 must be contained either in an elliptic paraboloid or in an affine image of the hypersurfacex 1 x 2...x n+1=const. We prove also that an affine complete, affine maximal surface is an elliptic paraboloid if its affine normals omit 4 or more directions in general position.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Calabi, E., Convex affine maximal surfaces,Results in Math.,13(1988), 199–223.
Cheng, S.Y. and Yau, S.T., Complete affine hypersurfaces, Part 1. The completeness of affine metrics,Common Pure and Appl. Math.,36(1986), 839–866.
Chern, S.S., Affine minimal hypersurfaces, Proc.Jap-U.S.Semin., Tokyo, 1977, 1988, 17–30.
Li Anmin, Uniqueness theorems in affine differential geometry. Part 1,Results in Mathematics,13(1988), 283–307.
Li Anmin and Gabi Penn, Uniqueness theorems in affine differential geometry. Part II,Results in Mathematics. 13(1988), 308–317.
Li Anmin, Affine maximal surfaces and harmonic functions, Lec. Notes,1369(1989),
Schneider, R., Zur affinen Differentialgeometrie im GroBen, 1,Math. Z.,101(1967), 375–406.
Schwenk, A. and Simon, U., Hypersurfaces with constant equiaffine mean curvature,Arch. Math.,46(1986), 85–90.
Author information
Authors and Affiliations
Additional information
The Project Supported by National Natural Science Foundation of China
Rights and permissions
About this article
Cite this article
Anmin, L. Some theorems in affine differential geometry. Acta Mathematica Sinica 5, 345–354 (1989). https://doi.org/10.1007/BF02107712
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02107712