Two results in the affine hypersurface theory
type: Item
instance of: scholarly article
title: Two results in the affine hypersurface theory
zbMATH Open document ID: 0684.53013
author: Takashi Kurose
published in: Journal of the Mathematical Society of Japan
publication date: 1989-01-01T00:00:00Z
review text: The first result gives a local classification of locally strongly convex affine spheres in affine 3-spaces with equiaffine constant curvature metric. This result was independently proved by \textit{A.-M. Li} and \textit{G. Penn} [Result. Math. 13, No.3/4, 308-317 (1988; Zbl 0646.53009)] and partly generalized to higher dimensions by \textit{A.-M. Li} [Acta Math. Sin., New Ser. 5, 345-354 (1989)] and \textit{J. H. Yu} [Affine hyperspheres with constant sectional curvature in \(A^ 4\) (Preprint Sichuan University, China)] for locally strongly convex hypersurfaces. Meanwhile all affine 2-spheres (also with indefinite metric) are classified [the reviewer, `Local classification of twodimensional affine spheres with constant curvature metric, Differ. Geom. Appl. (Brno) (to appear)], which is based on results of \textit{M. Magid} and \textit{P. Ryan} [Flat affine spheres in \({\mathbb{R}}^ 3\) (to appear)]. The second result is the affine version of the global existence and uniqueness of Minkowski for ovaloids. This affine-Euclidean relation was already known to Blaschke [\textit{W. Blaschke}, Gesammelte Werke. Vol. 4 (Essen 1985; Zbl 0656.53002), see the commentary there, p. 53 and p. 63 for further references]. A generalization of the uniqueness part can be found in [the reviewer, Recent developments in affine differential geometry, differential geometry and its applications, Proc. Conf., Dubrovnik/Yugosl. 1988, 327-347 (1989)].
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 53C45
zbMATH DE Number: 4120920
zbMATH Keywords: Minkowski problem
zbMATH Keywords: affine spheres
zbMATH Keywords: constant curvature
zbMATH Keywords: ovaloids
reviewed by: Udo Simon
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04130539
OpenAlex ID: W2107414616
Wikidata QID: Q125371833
DOI: 10.2969/JMSJ/04130539
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description: scientific article; zbMATH DE number 4120920