Two families of affine projectively flat surfaces
type: Item
instance of: scholarly article
title: Two families of affine projectively flat surfaces
zbMATH Open document ID: 0879.53011
DOI: 10.1007/BF01222932
author: Włodzimierz Jelonek
published in: Journal of Geometry
publication date: 1997-05-28T00:00:00Z
review text: In affine differential geometry, it follows from the basic existence and uniqueness theorem that the conjugate connection is projectively flat. Therefore, by assuming that also the induced connection is projectively flat, one obtains dual immersions with the same affine metric (but with the induced connection and conormal connection interchanged). In the present paper, the author studies affine surfaces whose induced connection is projectively flat and for which the shape operator is nondiagonalisable. Besides ruled surfaces with constant mean curvature, he obtains 3 special classes of affine surfaces. Surprisingly, all of the examples in these families are homogeneous with respect to the full affine group of transformations. The case of projectively flat surfaces with diagonalisable shape operator is considered in [\textit{F. Podesta}, Proc. Am. Math. Soc. 119, 255-260 (1993; Zbl 0790.53010)].
reviewed by: Luc Vrancken
Mathematics Subject Classification ID: 53A15
zbMATH DE Number: 1000350
zbMATH Keywords: projective flatness
zbMATH Keywords: affine surfaces
zbMATH Keywords: induced connection
zbMATH Keywords: shape operator
MaRDI profile type: Publication
Wikidata QID: Q125360615
cites work: Equivalence theorems in affine differential geometry
cites work: Conjugate connections and Radon's theorem in affine differential geometry
cites work: On the geometry of affine immersions
cites work: Projectively Flat Surfaces in  3
cites work: Affine surfaces with constant affine curvatures
description: scientific article; zbMATH DE number 1000350