Abstract
In this paper, we study locally strongly convex centroaffine hypersurfaces with parallel cubic form with respect to the Levi–Civita connection of the centroaffine metric. As the main result, we obtain a complete classification of such centroaffine hypersurfaces. The result of this paper is a centroaffine version of the complete classification of locally strongly convex equiaffine hypersurfaces with parallel cubic form due to Hu et al. (J Differ Geom 87:239–307, 2011).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Birembaux, O., Djorić, M.: Isotropic affine spheres. Acta Math. Sin. (Engl. Ser.) 28, 1955–1972 (2012)
Bokan, N., Nomizu, K., Simon, U.: Affine hypersurfaces with parallel cubic forms. Tôhoku Math. J. 42, 101–108 (1990)
Bott, R., Milnor, J.: On the parallelizability of the spheres. Bull. Am. Math. Soc. 64, 87–89 (1958)
Cheng, X.X., Hu, Z.J.: Classification of locally strongly convex isotropic centroaffine hypersurfaces, submitted. Preprint (2016)
Dillen, F., Vrancken, L.: 3-Dimensional affine hypersurfaces in \({\mathbb{R}}^{4}\) with parallel cubic form. Nagoya Math. J. 124, 41–53 (1991)
Dillen, F., Vrancken, L., Yaprak, S.: Affine hypersurfaces with parallel cubic form. Nagoya Math. J. 135, 153–164 (1994)
Dillen, F., Verbouwe, G., Vrancken, L.: Cubic form geometry for immersions in centro-affine and graph hypersurfacess. Results Math. 43, 88–95 (2003)
Hildebrand, R.: Centro-affine hypersurface immersions with parallel cubic form. Beitr. Algebra Geom. 56, 593–640 (2015)
Hu, Z.J., Li, C.C., Li, H., Vrancken, L.: Lorentzian affine hypersurfaces with parallel cubic form. Results Math. 59, 577–620 (2011)
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)
Hu, Z.J., Li, H., Vrancken, L.: Characterization of the Calabi product of hyperbolic affine hyperspheres. Results Math. 52, 299–314 (2008)
Hu, Z.J., Li, H., Vrancken, L.: Locally strongly convex affine hypersurfaces with parallel cubic form. J. Differ. Geom. 87, 239–307 (2011)
Kervaire, M.: Non-parallelizability of the \(n\)-sphere for \(n>7\). Proc. N. A. S. 44(3), 280–283 (1958)
Li, A.-M., Li, H., Simon, U.: Centroaffine Bernstein problems. Differ. Geom. Appl. 20, 331–356 (2004)
Li, A.-M., Liu, H.L., Schwenk-Schellschmidt, A., Simon, U., Wang, C.P.: Cubic form methods and relative Tchebychev hypersurfaces. Geom. Dedicata 66, 203–221 (1997)
Li, A.-M., Simon, U., Zhao, G., Hu, Z.J.: Global Affine Differential Geometry of Hypersurfaces, 2nd edn. W. de Gruyter, Berlin (2015)
Li, A.-M., Wang, C.P.: Canonical centroaffine hypersurfaces in \({\mathbb{R}}^{n+1}\). Results Math. 20, 660–681 (1991)
Liu, H.L., Wang, C.P.: The centroaffine Tchebychev operator. Results Math. 27, 77–92 (1995)
Liu, H.L., Wang, C.P.: Centroaffine surfaces with parallel traceless cubic form. Bull. Belg. Math. Soc. 4, 493–499 (1997)
Nomizu, K., Sasaki, T.: Affine Differential Geometry. Geometry of Affine Immersions. Cambridge Tracts in Mathematics, vol. 111. Cambridge University Press, Cambridge (1994)
Simon, U., Schwenk-Schellschmidt, A., Viesel, H.: Introduction to the Affine Differential Geometry of Hypersurfaces. Lecture Notes of the Science University of Tokyo. Science University of Tokyo, Tokyo (1991)
Vrancken, L., Li, A.-M., Simon, U.: Affine spheres with constant affine sectional curvature. Math. Z. 206, 651–658 (1991)
Wang, C.P.: Centroaffine minimal hypersurfaces in \({\mathbb{R}}^{n+1}\). Geom. Dedicata 51, 63–74 (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
This project was supported by NSFC (Grant No. 11371330).
Rights and permissions
About this article
Cite this article
Cheng, X., Hu, Z. & Moruz, M. Classification of the Locally Strongly Convex Centroaffine Hypersurfaces with Parallel Cubic Form. Results Math 72, 419–469 (2017). https://doi.org/10.1007/s00025-017-0651-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-017-0651-2
Keywords
- Centroaffine hypersurface
- locally strongly convex
- difference tensor
- Levi–Civita connection
- parallel cubic form