A differential geometric characterization of the Cayley hypersurface
HTML articles powered by AMS MathViewer
- by Zejun Hu, Cece Li and Dong Zhang PDF
- Proc. Amer. Math. Soc. 139 (2011), 3697-3706 Request permission
Abstract:
The so-called Cayley hypersurface, constructed by Eastwood and Ezhov, is a higher-dimensional extension of the classical Cayley surface. In this paper, we establish a differential geometric characterization of the Cayley hypersurface, which is an answer to Eastwood and Ezhov’s question.References
- Neda Bokan, Katsumi Nomizu, and Udo Simon, Affine hypersurfaces with parallel cubic forms, Tohoku Math. J. (2) 42 (1990), no. 1, 101–108. MR 1036477, DOI 10.2748/tmj/1178227697
- Yuncherl Choi and Hyuk Kim, A characterization of Cayley hypersurface and Eastwood and Ezhov conjecture, Internat. J. Math. 16 (2005), no. 8, 841–862. MR 2168070, DOI 10.1142/S0129167X05003168
- Franki Dillen and Luc Vrancken, $3$-dimensional affine hypersurfaces in $\textbf {R}^4$ with parallel cubic form, Nagoya Math. J. 124 (1991), 41–53. MR 1142975, DOI 10.1017/S0027763000003767
- Franki Dillen and Luc Vrancken, Generalized Cayley surfaces, Global differential geometry and global analysis (Berlin, 1990) Lecture Notes in Math., vol. 1481, Springer, Berlin, 1991, pp. 36–47. MR 1178516, DOI 10.1007/BFb0083626
- Franki Dillen and Luc Vrancken, Hypersurfaces with parallel difference tensor, Japan. J. Math. (N.S.) 24 (1998), no. 1, 43–60. MR 1630113, DOI 10.4099/math1924.24.43
- Franki Dillen, Luc Vrancken, and Sahnur Yaprak, Affine hypersurfaces with parallel cubic form, Nagoya Math. J. 135 (1994), 153–164. MR 1295822, DOI 10.1017/S0027763000005006
- M. Eastwood and V. Ezhov, Cayley hypersurfaces, Tr. Mat. Inst. Steklova 253 (2006), no. Kompleks. Anal. i Prilozh., 241–244; English transl., Proc. Steklov Inst. Math. 2(253) (2006), 221–224. MR 2338700, DOI 10.1134/s0081543806020180
- Z.J. Hu and C.C. Li, The classification of $3$-dimensional Lorentzian affine hypersurfaces with parallel cubic form, preprint (2010).
- Zejun Hu, Haizhong Li, Udo Simon, and Luc Vrancken, On locally strongly convex affine hypersurfaces with parallel cubic form. I, Differential Geom. Appl. 27 (2009), no. 2, 188–205. MR 2503972, DOI 10.1016/j.difgeo.2008.10.005
- Zejun Hu, Haizhong Li, and Luc Vrancken, Characterizations of the Calabi product of hyperbolic affine hyperspheres, Results Math. 52 (2008), no. 3-4, 299–314. MR 2443493, DOI 10.1007/s00025-008-0312-6
- Z.J. Hu, H. Li and L. Vrancken, Locally strongly convex affine hypersurfaces with parallel cubic form, J. Diff. Geom., to appear.
- Martin A. Magid and Katsumi Nomizu, On affine surfaces whose cubic forms are parallel relative to the affine metric, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 7, 215–218. MR 1030183
- Katsumi Nomizu and Ulrich Pinkall, Cayley surfaces in affine differential geometry, Tohoku Math. J. (2) 41 (1989), no. 4, 589–596. MR 1025324, DOI 10.2748/tmj/1178227729
- Katsumi Nomizu and Takeshi Sasaki, Affine differential geometry, Cambridge Tracts in Mathematics, vol. 111, Cambridge University Press, Cambridge, 1994. Geometry of affine immersions. MR 1311248
- Luc Vrancken, Affine higher order parallel hypersurfaces, Ann. Fac. Sci. Toulouse Math. (5) 9 (1988), no. 3, 341–353 (English, with English and French summaries). MR 1425270
- Luc Vrancken, The Magid-Ryan conjecture for equiaffine hyperspheres with constant sectional curvature, J. Differential Geom. 54 (2000), no. 1, 99–138. MR 1815413
Additional Information
- Zejun Hu
- Affiliation: Department of Mathematics, Zhengzhou University, Zhengzhou 450052, People’s Republic of China
- MR Author ID: 346519
- ORCID: 0000-0003-2744-5803
- Email: huzj@zzu.edu.cn
- Cece Li
- Affiliation: Department of Mathematics, Zhengzhou University, Zhengzhou 450052, People’s Republic of China
- Email: ceceli@sina.com
- Dong Zhang
- Affiliation: Department of Mathematics, Zhengzhou University, Zhengzhou 450052, People’s Republic of China
- Email: zd20082100333@163.com
- Received by editor(s): April 29, 2010
- Received by editor(s) in revised form: August 26, 2010
- Published electronically: February 21, 2011
- Additional Notes: This project was supported by grants of the NSFC (No. 10671181 and No. 11071225)
- Communicated by: Jianguo Cao
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 3697-3706
- MSC (2010): Primary 53A15; Secondary 53B25, 53B30
- DOI: https://doi.org/10.1090/S0002-9939-2011-10772-X
- MathSciNet review: 2813399