Hypersurfaces with parallel cubic form in affine space
type: Item
instance of: scholarly article
title: Hypersurfaces with parallel cubic form in affine space
zbMATH Open document ID: 0822.53005
author: A. V. Chakmazyan
published in: Russian Mathematics
publication date: 1995-05-17T00:00:00Z
review text: Let \(M_{n-1}\) be a hypersurface in an affine \(n\)-dimensional space \(A_ n\), and let \(\nabla\), \(\nabla^ \perp\) and \(\overline{\nabla} = \nabla \oplus \nabla^ \perp\) be connections in the tangent bundle, the normal bundle and the van der Waerden-Bortolotti connection, respectively. If \(\alpha_ 2\) is the second fundamental form of the hypersurface \(M_{n-1}\), then its third fundamental form \(\alpha_ 3 = \overline{\nabla} \alpha_ 2\). If \(\overline {\nabla} \alpha_ 3 = 0\), then the hypersurface \(M_{n-1}\) is said to have parallel third fundamental form. In the paper under review, the author finds a local structure of a tangentially nondegenerate hypersurface \(M_{n-1}\) with parallel third fundamental form with respect to the Blaschke normals. He proves that locally such a hypersurface coincides with a certain improper affine hypersphere.
Mathematics Subject Classification ID: 53A15
zbMATH DE Number: 746049
zbMATH Keywords: \(n\)-dimensional affine space
zbMATH Keywords: van der Waerden-Bortolotti connection
zbMATH Keywords: parallel third fundamental form
zbMATH Keywords: improper affine hypersphere
reviewed by: Vladislav V. Goldberg
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 746049