Skip to main content
SpringerLink
Log in

Multiple product affine hypersurfaces with semi-parallel cubic form

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

In this paper, we study the multiple Riemannian products \({\mathbb {R}}^q\times \prod _{i=1}^{r}M_i^{n_i}(c_i)\), immersed in the affine space as locally strongly convex affine hypersurfaces with semi-parallel cubic form relative to the Levi-Civita connection of affine metric, where each \(M_i^{n_i}(c_i)\) is a space form of nonzero curvature \(c_i\). As the main result, we establish the classification of such hypersurfaces if either \(q\le r\) or the immersions being affine hyperspheres.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Antić, M., Dillen, F., Schoels, K., Vrancken, L.: Decomposable affine hypersurfaces. Kyushu J. Math. 68, 93–103 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Antić, M., Hu, Z., Li, C., Vrancken, L.: Characterization of the generalized Calabi composition of affine hyperspheres. Acta. Math. Sin. (Engl. Ser.) 31, 1531–1554 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Antić, M., Li, H., Vrancken, L., Wang, X.: Affine hypersurfaces with constant sectional curvature. Pac. J. Math. 310, 275–302 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  4. Birembaux, O., Djorić, M.: Isotropic affine spheres. Acta. Math. Sin. (Engl. Ser.) 28, 1955–1972 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brozos-Vázquez, M., García-Ró, E., Vázquez-Lorenzo, R.: Complete locally conformally flat manifolds of negative curvature. Pacific J. Math. 226, 201–219 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Birembaux, O., Vrancken, L.: Isotropic affine hypersurfaces of dimension \(5\). J. Math. Anal. Appl. 417, 918–962 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Calabi, E.: Complete affine hyperspheres. I. Sympos. Math. 10, 19–38 (1972)

    MathSciNet  MATH  Google Scholar 

  9. Cheng, X., Hu, Z., Li, A.-M., Li, H.: On the isolation phenomena of Einstein manifolds-submanifolds versions. Proc. Am. Math. Soc. 146, 1731–1740 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cheng, X., Hu, Z., Moruz, M., Vrancken, L.: On product affine hyperspheres in \({\mathbb{R} }^{n+1}\). Sci. China Math. 63, 2055–2078 (2020)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Dillen, F., Vrancken, L.: Calabi-type composition of affine spheres. Diff. Geom. Appl. 4, 303–328 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. 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  MATH  Google Scholar 

  16. Hildebrand, R.: Centro-affine hypersurface immersions with parallel cubic form. Beitr. Algebra Geom. 56, 593–640 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hildebrand, R.: Graph immersions with parallel cubic form. Diff. Geom. Appl. 74, 101700 (2021)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Hu, Z., Li, 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Hu, Z., Li, H., Vrancken, L.: Characterizations of the Calabi product of hyperbolic affine hyperspheres. Results Math. 52, 299–314 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  24. Hu, Z., Xing, C.: New equiaffine characterizations of the ellipsoids related to an equiaffine integral inequality on hyperovaloids. Math. Inequal. Appl. 24, 337–350 (2021)

    MathSciNet  MATH  Google Scholar 

  25. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. I. Interscience Publishers, New York (1963)

    MATH  Google Scholar 

  26. Li, A.-M., Simon, U., Zhao, G., Hu, Z.: Global Affine Differential Geometry of Hypersurfaces, 2 nd ed. Walter de Gruyter, Berlin (2015)

  27. Li, C., Xing, C., Xu, H.: Locally strongly convex affine hypersurfaces with semi-parallel cubic form. J. Geom. Anal. 33, 81 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  28. Li, C., Zhang, D.: The generalized Cayley hypersurfaces and their geometrical characterization. Results Math. 68, 25–44 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li, X.: A new characterization of Calabi composition of hyperbolic affine hyperspheres. Results Math. 66, 137–158 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. 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  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  33. O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1983)

    MATH  Google Scholar 

  34. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Xu, H., Li, C.: Conformally flat affine hypersurfaces with semi-parallel cubic form. Acta Math. Sci. in press (2023) arXiv:2211.02814

Download references

Acknowledgements

The authors would like to thank the referees for their valuable comments and pointing several misprints which make the paper more readable.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471000, People’s Republic of China

    Cece Li & Huiyang Xu

Authors
  1. Cece Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huiyang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cece Li.

Ethics declarations

Conflict of interest

The authors have no competing interests to declare that are relevant to the content of this article.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported in part by National Natural Science Foundation of China (Grant Numbers 12101194, 11401173).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, C., Xu, H. Multiple product affine hypersurfaces with semi-parallel cubic form. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 123 (2023). https://doi.org/10.1007/s13398-023-01455-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-023-01455-1

Keywords

Mathematics Subject Classification

Search

Navigation