A differential geometric characterization of the Cayley hypersurface

Authors:
Zejun Hu, Cece Li and Dong Zhang

Journal:
Proc. Amer. Math. Soc. **139** (2011), 3697-3706

MSC (2010):
Primary 53A15; Secondary 53B25, 53B30

Published electronically:
February 21, 2011

MathSciNet review:
2813399

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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.

**1.**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**, 10.2748/tmj/1178227697**2.**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**, 10.1142/S0129167X05003168**3.**Franki Dillen and Luc Vrancken,*3-dimensional affine hypersurfaces in 𝑅⁴ with parallel cubic form*, Nagoya Math. J.**124**(1991), 41–53. MR**1142975****4.**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**, 10.1007/BFb0083626**5.**Franki Dillen and Luc Vrancken,*Hypersurfaces with parallel difference tensor*, Japan. J. Math. (N.S.)**24**(1998), no. 1, 43–60. MR**1630113****6.**Franki Dillen, Luc Vrancken, and Sahnur Yaprak,*Affine hypersurfaces with parallel cubic form*, Nagoya Math. J.**135**(1994), 153–164. MR**1295822****7.**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****8.**Z.J. Hu and C.C. Li,*The classification of -dimensional Lorentzian affine hypersurfaces with parallel cubic form*, preprint (2010).**9.**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**, 10.1016/j.difgeo.2008.10.005**10.**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**, 10.1007/s00025-008-0312-6**11.**Z.J. Hu, H. Li and L. Vrancken,*Locally strongly convex affine hypersurfaces with parallel cubic form*, J. Diff. Geom., to appear.**12.**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****13.**Katsumi Nomizu and Ulrich Pinkall,*Cayley surfaces in affine differential geometry*, Tohoku Math. J. (2)**41**(1989), no. 4, 589–596. MR**1025324**, 10.2748/tmj/1178227729**14.**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****15.**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****16.**Luc Vrancken,*The Magid-Ryan conjecture for equiaffine hyperspheres with constant sectional curvature*, J. Differential Geom.**54**(2000), no. 1, 99–138. MR**1815413**

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Additional Information

**Zejun Hu**

Affiliation:
Department of Mathematics, Zhengzhou University, Zhengzhou 450052, People’s Republic of China

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

DOI:
https://doi.org/10.1090/S0002-9939-2011-10772-X

Keywords:
Cayley hypersurface,
differential geometric characterization,
parallel cubic form

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

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.