Parallel affine immersions \(M^n\to\mathbb R^{n+2}\) with flat connection
type: Item
instance of: scholarly article
title: Parallel affine immersions \(M^n\to\mathbb R^{n+2}\) with flat connection
zbMATH Open document ID: 1341.53017
author: O. O. Shugailo
published in: Ukrainian Mathematical Journal
publication date: 2015-11-13T00:00:00Z
review text: An affine immersion is called \textit{parallel}, if the cubic form vanishes identically. The paper under review is devoted to the study of parallel affine immersions \(f:M^n\rightarrow \mathbb R^{n+2}\) with flat connection. The so called \textit{point codimension \(q\)} is defined as the rank of the affine fundamental form and does not depend on the choice of the transverse distribution. The cases possible are \(q=0,1,2\). While the first one gives an affine subspace of \(\mathbb R^n\), the second one leads to the hypersurface situation and thus to hyperquadrics. So the main discussion is the solution in case \(q=2\), where a complete classification is given, depending on the rank of the Weingarten mapping.
reviewed by: Friedrich Manhart
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 53A07
zbMATH DE Number: 6508314
zbMATH Keywords: parallel affine immersion
zbMATH Keywords: codimension-2 immersions
zbMATH Keywords: flat connection
MaRDI profile type: Publication
full work available at URL: s11253-014-0870-6
OpenAlex ID: W1980569452
cites work: Q4846450
cites work: Q4510897
cites work: Immersions with parallel second fundamental form
cites work: Parallel affine immersions with maximal codimension
cites work: Parallel surfaces in affine 4-space
cites work: On the geometry of affine immersions
cites work: A NEW EQUIAFFINE THEORY FOR SURFACES IN ℝ4
cites work: Affine surfaces in \(\mathbb{R}^5\) with zero cubic form
cites work: Q3837232
cites work: On affine immersions with flat connections
cites work: On affine hypersurfaces with parallel nullity
cites work: Affine submanifolds of rank two
cites work: Flat affine surfaces in \(\mathbb{R}^4\) with flat normal connection
Wikidata QID: Q125927151
DOI: 10.1007/S11253-014-0870-6
description: scientific article; zbMATH DE number 6508314