Classification of locally strongly convex isotropic centroaffine hypersurfaces

Cheng, Xiuxiu; Hu, Zejun

doi:10.1016/j.difgeo.2019.03.004

Cited by 12 publications

(3 citation statements)

References 21 publications

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“…Remark 1.3. Related to Theorem 1.1 we have established in [4] the classification of locally strongly convex isotropic centroaffine hypersurfaces. From the comparison of the main results in [1] and [4] one sees that the isotropic condition again have different implications in both equiaffine theory of hypersurfaces and centroaffine theory of hypersurfaces, just like Theorem 1.1 here and the Classification Theorem in [12].…”

Section: Introductionmentioning

confidence: 96%

Classification of the Locally Strongly Convex Centroaffine Hypersurfaces with Parallel Cubic Form

Cheng

Moruz

2017

Results Math

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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, Li and Vrancken [12].

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Section: Introductionmentioning

confidence: 96%

Classification of the Locally Strongly Convex Centroaffine Hypersurfaces with Parallel Cubic Form

Cheng

Moruz

2017

Results Math

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“…Remark 1.3. Besides that as stated in [4], different characterizations on the typical examples of centroaffine hypersurfaces appearing in Theorem 1.1 were established in our recent articles, [2] and [3], from other aspects of differential geometric invariants.…”

Section: Introductionmentioning

confidence: 99%

An Optimal Inequality on Locally Strongly Convex Centroaffine Hypersurfaces

Cheng

2017

J Geom Anal

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Abstract. In this paper, we establish a general inequality for locally strongly convex centroaffine hypersurfaces in R n+1 involving the norm of the covariant derivatives of both the difference tensor K and the Tchebychev vector field T . Our result is optimal in that, applying our recent classification for locally strongly convex centroaffine hypersurfaces with parallel cubic form in [4], we can completely classify the hypersurfaces which realize the equality case of the inequality.

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“…In affine differential geometry, hypersurfaces with isotropic difference tensor K have been studied in [2,4,14]. For centroaffine hypersurfaces, X. X. Cheng and Z. J. Hu classified locally strongly convex isotropic centroaffine hypersurfaces for all dimensions in [6]. Olivier Birembaux [1] studied lightlike isotropic but not pseudo-isotropic Lorentzian centroaffine surfaces M 2 in R 3 and classified all those surfaces.…”

Section: Introductionmentioning

confidence: 99%