Classification of the Conformally Flat Centroaffine Hypersurfaces with Vanishing Centroaffine Shape Operator

Abstract

Cheng-Hu-Moruz (2017) completely classified the locally strongly convex centroaffine hypersurfaces with parallel cubic form based on the Calabi product (called the type I Calabi product for short) proposed by Li-Wang (1991).

In the present paper, the authors introduce the type II Calabi product (in case λ₁ = 2λ₂), complementing the type I Calabi product (in case λ₁ ≠ 2λ₂), and achieve a classification of the locally strongly convex centroaffine hypersurfaces in ℝⁿ⁺¹ with vanishing centroaffine shape operator and Weyl curvature tensor by virtue of the types I and II Calabi product.

As a corollary, 3-dimensional complete locally strongly convex centroaffine hypersurfaces with vanishing centroaffine shape operator are completely classified, which positively answers the centroaffine Bernstein problems III and V by Li-Li-Simon (2004).

Keywords

Centroaffine hypersurface / Centroaffine shape operator / Calabi product / Locally conformally flat / Calabi hypersurface

References

[1] BinderT. Relative Tchebychev hypersurfaces which are also translation hypersurfaces. Hokkaido Math. J., 2009, 38(1): 97-110

[2] CalabiE. Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens. Michigan Math. J., 1958, 5(2): 105-126

[3] ChengX X, HuZ J, MoruzM. Classification of the locally strongly convex centroaffine hypersurfaces with parallel cubic form. Results Math., 2017, 72(1–2): 419-469

[4] ChengX X, HuZ J, MoruzM, VranckenL. On product affine hyperspheres in ℝⁿ⁺¹. Sci. China Math., 2020, 63(10): 2055-2078

[5] ChengX X, HuZ J, VranckenL. Every centroaffine Tchebychev hyperovaloid is ellipsoid. Pacific J. Math., 2021, 315(1): 27-44

[6] ChengX X, HuZ J, XingC. On centroaffine Tchebychev hypersurfaces with constant sectional curvature. Results Math., 2022, 77(4): 17529pp

[7] ChengX X, HuZ J, YaoZ K. A rigidity theorem for centroaffine Chebyshev hyperovaloids. Colloq. Math., 2019, 157(1): 133-141

[8] HuZ J, XingC. Locally conformally flat affine hyperspheres with parallel Ricci tensor. J. Math. Anal. Appl., 2023, 528(1): 12759611pp

[9] LeiM X, XuR W. Classification of Calabi hypersurfaces in ℝⁿ⁺¹ with parallel Fubini-Pick tensor. J. Geom. Anal., 2024, 34(6): 16930pp

[10] LiA-M, LiH Z, SimonU. Centroaffine Bernstein problems. Differential Geom. Appl., 2004, 20(3): 331-356

[11] LiA-M, LiuH L, Schwenk-SchellschmidtA, et al.. Cubic form methods and relative Tchebychev hypersurfaces. Geom. Dedicata, 1997, 66(2): 203-221

[12] LiA-M, SimonU, ZhaoG S, HuZ JGlobal Affine Differential Geometry of Hypersurfaces, 2015, Berlin, De GruyterSecond revised and extended edition

[13] LiA-M, WangC P. Canonical centroaffine hypersurfaces in ℝⁿ⁺¹. Results Math., 1991, 20(3–4): 660-681

[14] LiuH L, SimonU, WangC P. Conformal structure in affine geometry: Complete Tchebychev hypersurfaces. Abh. Math. Semin. Univ. Hambg., 1996, 66: 249-262

[15] LiuH L, WangC P. The centroaffine Tchebychev operator. Results Math., 1995, 27(1–2): 77-92

[16] LiuH L, WangC P. Relative Tchebychev surfaces in ℝ³. Kyushu J. Math., 1996, 50(2): 533-540

[17] NomizuK, SasakiTAffine Differential Geometry, Geometry of Affine Immersions, 1994, Cambridge, Cambridge University Press

[18] O’NeillB. Semi-Riemannian Geometry. With Applications to Relativity, 1983, New York, Academic Press, Inc.

[19] PogorelovA VThe Minkowski Multidimensional Problem, 1978, Washington, DC, V. H. Winston & SonsHalsted Press, John Wiley & Sons, New York-Toronto-London

[20] SimonU, Schwenk-SchellschmidtA, VieselHIntroduction to the Affine Differential Geometry of Hypersurfaces, 1991, Tokyo, Science University of Tokyo

[21] SimonU, TrabelsiH, VranckenL. Complete hyperbolic Tchebychev hypersurfaces. J. Geom., 2008, 89(1–2): 148-159

[22] WangC P. Centroaffine minimal hypersurfaces in ℝⁿ⁺¹. Geom. Dedicata, 1994, 51(1): 63-74

[23] XuH Y, LiC C. On conformally flat affine hypersurfaces with pseudo-parallel cubic form. J. Geom. Phys., 2023, 187: 10477812pp

[24] XuR W, LeiM X. Classification of Calabi hypersurfaces with parallel Fubini-Pick form. Differential Geom. Appl., 2021, 74: 10170716pp

[25] XuR W, LiX X. On the complete solutions to the Tchebychev affine Kähler equation and its geometric significance. J. Math. Anal. Appl., 2022, 514(2): 12635118pp

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