Immersions of the real projective plane into \(\mathbb R^{3}\)
type: Item
instance of: scholarly article
title: Immersions of the real projective plane into \(\mathbb R^{3}\)
zbMATH Open document ID: 1231.53009
author: François Apéry
published in: Mathematische Semesterberichte
publication date: 2011-05-25T00:00:00Z
review text: Several immersions of the real projective plane \({\mathbb P}^2\) into \({\mathbb R}^3\) are discussed. A classical way to obtain such immersions is to use the Veronese embedding \((x:y:z) \longrightarrow (x^2:xy:xz:y^2:yz:z^2)\) of \({\mathbb P}^2\) into \({\mathbb R}^5\) followed by a projection into \({\mathbb R}^3\) whose center is a straight line. Such representations of \({\mathbb P}^2\) as surfaces in \({\mathbb R}^3\) are commonly known under the name Steiner surfaces. Prominent examples are the cross cap and Jakob Steiner's Roman surface. Parameterizations and singularities of such surfaces are discussed in this paper. Another way to obtain immersions of \({\mathbb P}^2\) into \({\mathbb R}^3\) is presented by the author. He uses a family of mappings \(f_{2m+1}\) from the 2-sphere \({\mathbb S}^2\) into \({\mathbb R}^3\), where \(m\in {\mathbb N}\). The surface \(\Sigma_{2m+1}\) defined via such a mapping is an immersion of \({\mathbb P}^2\) into \({\mathbb R}^3\) being algebraic of degree \(4m+2\) and displaying \((2m+1)\)-fold axial symmetry. In the case of \(m=1\), one gets the classical surface of Werner Boy. It is also shown how Boy's surface can be obtained as a projection of an embedding of \({\mathbb P}^2\) in \({\mathbb R}^5\).
reviewed by: Anton Gfrerrer
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 57R42
zbMATH DE Number: 5899614
zbMATH Keywords: real projective plane
zbMATH Keywords: immersion
zbMATH Keywords: Boy's surface
zbMATH Keywords: Steiner surface
zbMATH Keywords: Roman surface
zbMATH Keywords: cross cap
MaRDI profile type: Publication
full work available at URL: s00591-010-0079-1
OpenAlex ID: W1967050738
cites work: La surface de Boy. (Boy's surface)
cites work: Q3761286
cites work: An algebraic halfway model for the eversion of the sphere (with an appendix by Bernard Morin)
cites work: Q4046802
cites work: Q5620217
cites work: Q3672755
DOI: 10.1007/S00591-010-0079-1
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description: scientific article; zbMATH DE number 5899614