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The Generalized Cayley Hypersurfaces and Their Geometrical Characterization

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Abstract

In this paper, we study a whole family of n-dimensional equiaffine homogeneous hypersurfaces with a parameter α, constructed by Eastwood and Ezhov (Proc Steklov Inst Math 253:221–224, 2006), called the generalized Cayley hypersurfaces. By introducing a new parametrization we find that the generalized Cayley hypersurfaces are improper affine hypersphere with flat affine metric and vanishing Pick invariant, whose difference tensor K satisfies \({\nabla^{(\alpha)}K=0 \,\,{\rm and}\,\, K^{n-1} \neq 0}\), where the affine \({\alpha{\rm -connection}\,\, \nabla^{(\alpha)}}\) of information geometry is first introduced on affine hypersurface for each \({\alpha\in\mathbb{R}}\). As main result, we establish a characterization of the generalized Cayley hypersurfaces by the last two properties for some nonzero constant α.

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References

  1. Bokan N., Nomizu K., Simon U.: Affine hypersurfaces with parallel cubic forms. Tôhoku Math. J. 42, 101–108 (1990)

    Article  MathSciNet  Google Scholar 

  2. Brualdi R.A.: Introductory Combinatorics. Prentice Hall, Upper Saddle River (1999)

    Google Scholar 

  3. Choi Y., Kim H.: A characterization of Cayley hypersurface and Eastwood and Ezhov conjecture. Int. J. Math. 16, 841–862 (2005)

    Article  Google Scholar 

  4. Dillen, F., Vrancken, L.: Generalized Cayley Surfaces. Lecture Notes in Mathematics, vol. 1481, pp. 36–47 (1990)

  5. Dillen F., Vrancken L.: 3-dimensional affine hypersurfaces in \({\mathbb{R}^4}\) with parallel cubic form. Nagoya Math. J. 124, 41–53 (1991)

    MathSciNet  Google Scholar 

  6. Dillen F., Vrancken L.: Hypersurfaces with parallel difference tensor. Jpn. J. Math. 24, 43–60 (1998)

    MathSciNet  MATH  Google Scholar 

  7. Dillen F., Vrancken L., Yaprak S.: Affine hypersurfaces with parallel cubic form. Nagoya Math. J. 135, 153–164 (1994)

    MathSciNet  MATH  Google Scholar 

  8. Eastwood M., Ezhov V.: Cayley hypersurfaces. Proc. Steklov Inst. Math. 253, 221–224 (2006)

    Article  MathSciNet  Google Scholar 

  9. Gigena S.: On affine hypersurfaces with parallel second fundamental form. Tôhoku Math. J. 54, 495–512 (2002)

    Article  MathSciNet  Google Scholar 

  10. Gigena S.: Classification of five dimensional hypersurfaces with affine normal parallel cubic form. Beitr. Algebra Geom. 44, 511–524 (2003)

    MathSciNet  Google Scholar 

  11. Gigena S.: Inductive schemes for the complete classification of affine hypersurfaces with parallel second fundamental form. Beitr. Algebra Geom. 52, 51–73 (2011)

    Article  MathSciNet  Google Scholar 

  12. Hu Z.J., Li C.C.: The classification of 3-dimensional Lorentzian affine hypersurfaces with parallel cubic form. Differ. Geom. Appl. 29, 361–373 (2011)

    Article  MATH  Google Scholar 

  13. Hu Z.J., Li H., Simon U., Vrancken L.: On locally strongly convex affine hypersurfaces with parallel cubic form. Part I. Differ. Geom. Appl. 27, 188–205 (2009)

    Article  MathSciNet  Google Scholar 

  14. Hu Z.J., Li C.C., Li H., Vrancken L.: Lorentzian affine hypersurfaces with parallel cubic form. Results Math. 59, 577–620 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hu Z.J., Li C.C., Li H., Vrancken L.: The classification of 4-dimensional non-degenerate affine hypersurfaces with parallel cubic form. J. Geom. Phys. 61, 2035–2057 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hu Z.J., Li H., Vrancken L.: Locally strongly convex affine hypersurfaces with parallel cubic form. J. Differ. Geom. 87, 239–307 (2011)

    MathSciNet  Google Scholar 

  17. Hu Z.J., Li C.C., Zhang D.: A differential geometry characterization of the Cayley hypersurface. Proc. Am. Math. Soc. 139, 3697–3706 (2011)

    Article  MathSciNet  Google Scholar 

  18. Li, C.C.: Affine hypersurfaces with parallel difference tensor relative to affine α-connection. J. Geom. Phys. 86, 81–93 (2014)

  19. Li A.M., Wang C.P.: Canonical centroaffine hypersurfaces in \({\mathbb{R}^{n+1}}\). Results Math. 20, 660–681 (1991)

    Article  MathSciNet  Google Scholar 

  20. Li, A.M., Simon, U., Zhao, G.S.: Global Affine Differential Geometry of Hypersurfaces. W. de Gruyter, Berlin (1993)

  21. Liu H.L., Wang C.P.: Centroaffine surfaces with parallel traceless cubic form. Bull. Belg. Math. Soc. 4, 493–499 (1997)

    Google Scholar 

  22. Magid M., Nomizu K.: On affine surfaces whose cubic forms are parallel relative to the affine metric. Proc. Jpn. Acad. Ser. A 65, 215–218 (1989)

    Article  MathSciNet  Google Scholar 

  23. Nomizu K., Pinkall U.: Cayley surfaces in affine differential geometry. Tôhoku Math. J. 41, 589–596 (1989)

    Article  MathSciNet  Google Scholar 

  24. Nomizu K., Sasaki T.: Affine Differential Geometry: Geometry of Affine Immersions. Cambridge University Press, Cambridge (1994)

    Google Scholar 

  25. Shima, H.: The Geometry of Hessian Structures. World Scientific, Singapore (2007)

  26. Vrancken L.: Affine higher order parallel hypersurfaces. Ann. Fac. Sci. Toulouse Math. 9, 341–353 (1988)

    Article  MathSciNet  Google Scholar 

  27. Vrancken L.: The Magid–Ryan conjecture for equiaffine hyperspheres with constant sectional curvature. J. Differ. Geom. 54, 99–138 (2000)

    MathSciNet  Google Scholar 

  28. Vrancken L., Li A.M., Simon U.: Affine spheres with constant affine sectional curvature. Math. Z. 206, 651–658 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang C.P.: Canonical equiaffine hypersurfaces in \({\mathbb{R}^{n+1}}\). Math. Z. 214, 579–592 (1993)

    Article  MathSciNet  Google Scholar 

  30. Zhang J.: A note on curvature of α-connections of a statistical manifold. AISM 59, 161–170 (2007)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471023, People’s Republic of China

    Cece Li

  2. School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

    Dong Zhang

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  1. Cece Li
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  2. Dong Zhang
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Correspondence to Cece Li.

Additional information

This research was supported by the Grants of NSFC-11326072 and NSFC-11401173.

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Li, C., Zhang, D. The Generalized Cayley Hypersurfaces and Their Geometrical Characterization. Results. Math. 68, 25–44 (2015). https://doi.org/10.1007/s00025-014-0419-x

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  • DOI: https://doi.org/10.1007/s00025-014-0419-x

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