Abstract
In this paper, we study the interesting open problem of classifying the locally strongly convex centroaffine Tchebychev hypersurfaces in \({\mathbb {R}}^{n+1}\) with constant sectional curvature. First, for arbitrary dimensions we solve the problem by assuming that the centroaffine shape operator vanishes. Second, extending the solved cases of \(n=2,3\), we continue working with the case \(n=4\). As the result, we establish a complete classification of the flat hyperbolic centroaffine Tchebychev hypersurfaces in \({\mathbb {R}}^5\).
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References
Binder, Th.: Relative Tchebychev hypersurfaces which are also translation hypersurfaces. Hokkaido Math. J. 38, 97–110 (2009)
Cheng, X., Hu, Z.: On the isolation phenomena of locally conformally flat manifolds with constant scalar curvature-submanifolds versions. J. Math. Anal. Appl. 464, 1147–1157 (2018)
Cheng, X., Hu, Z., Moruz, M.: Classification of the locally strongly convex centroaffine hypersurfaces with parallel cubic form. Results Math. 72, 419–469 (2017)
Cheng, X., Hu, Z., Moruz, M., Vrancken, L.: On product affine hyperspheres in \({\mathbb{R} }^{n+1}\). Sci China Math 63, 2055–2078 (2020)
Cheng, X., Hu, Z., Vrancken, L.: Every centroaffine Tchebychev hyperovaloid is ellipsoid. Pacific J. Math. 315, 27–44 (2021)
Cheng, X., Hu, Z., Yao, Z.: A rigidity theorem for centroaffine Chebyshev hyperovaloids. Colloq. Math. 157, 133–141 (2019)
Hu, Z., Li, H., Vrancken, L.: Locally strongly convex affine hypersurfaces with parallel cubic form. J. Differ. Geom. 87, 239–307 (2011)
Hu, Z., Xing, C.: A new centroaffine characterization of the ellipsoids. Proc. Amer. Math. Soc. 149, 3531–3540 (2021)
Li, M.: Rigidity theorems for relative Tchebychev hypersurfaces. Results Math. 70, 283–298 (2016)
Li, A.-M., Li, H., Simon, U.: Centroaffine Bernstein problems. Differ. Geom. Appl. 20, 331–356 (2004)
Li, A.-M., Liu, H., Schwenk-Schellschmidt, A., Simon, U., Wang, C.P.: Cubic form methods and relative Tchebychev hypersurfaces. Geom. Dedicata. 66, 203–221 (1997)
Li, A.-M., Penn, G.: Uniqueness theorems in affine differential geometry part II. Results Math. 13, 308–317 (1988)
Lalléchère, S., Ramifidisoa, L., Ravelo, B.: Flat hyperbolic centro-affine Tchebychev hypersurfaces of \(\mathbb{R}^4\). Results Math. 76: 71, 18 pp. (2021)
Li, A.-M., Simon, U., Zhao, G., Hu, Z.: Global Affine Differential Geometry of Hypersurfaces, 2nd edn. de Gruyter Expositions in Mathematics 11, Walter de Gruyter, Berlin/Boston (2015)
Liu, H., Simon, U., Wang, C.P.: Conformal structures in affine geometry: complete Tchebychev hypersurfaces. Abh. Math. Sem. Hamburg 66, 249–262 (1996)
Li, A.-M., Wang, C.P.: Canonical centroaffine hypersurfaces in \({\mathbb{R} }^{n+1}\). Results Math. 20, 660–681 (1991)
Liu, H., Wang, C.P.: The centroaffine Tchebychev operator. Results Math. 27, 77–92 (1995)
Liu, H., Wang, C.P.: Relative Tchebychev surfaces in \({\mathbb{R} }^3\). Kyushu J. Math. 50, 533–540 (1996)
Nomizu, K., Sasaki, T.: Affine Differential Geometry. 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, Japan (1991)
Simon, U., Trabelsi, H., Vrancken, L.: Complete hyperbolic Tchebychev hypersurfaces. J. Geom. 89, 148–159 (2008)
Vrancken, L.: The Magid-Ryan conjecture for equiaffine hyperspheres with constant sectional curvature. J. Differ. Geom. 54, 99–138 (2000)
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)
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The authors are thankful to the referees for their comments and suggestions which have helped us to improve the presentation of the article.
Funding
The first author was supported by NSF of China, Grant Number 12001494; the second and third authors were supported by NSF of China, Grant Number 12171437.
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Cheng, X., Hu, Z. & Xing, C. On Centroaffine Tchebychev Hypersurfaces with Constant Sectional Curvature. Results Math 77, 175 (2022). https://doi.org/10.1007/s00025-022-01715-3
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DOI: https://doi.org/10.1007/s00025-022-01715-3