Invariants of conformal and projective structures
type: Item
instance of: scholarly article
title: Invariants of conformal and projective structures
zbMATH Open document ID: 0838.53017
DOI: 10.1007/BF03322280
author: Christian Steglich
published in: Results in Mathematics
publication date: 1996-06-04T00:00:00Z
review text: Given a non-degenerate affine hypersurface \(M^n\) of \(\mathbb{R}^{n+ 1}\), then any so-called relative normalization (that is, the choice of a transversal vector field with vanishing transversal connection form) induces a semi-Riemannian metric \(h\) and a torsion-free Ricci-symmetric connection \(\nabla\) on \(M^n\). Then \(h\) and \(\nabla\) determine the conjugate connection \(\overline\nabla\), which is again torsion-free and Ricci-symmetric (in fact, the author always tacitly assumes connections to be torsion-free and Ricci-symmetric). When changing the relative normalization, these data obviously change, \(h\) by a conformal transformation, \(\overline\nabla\) by a strongly projective transformation and \(\nabla\) by a so-called conprojective tarnsformation. In this way one has a natural class of conjugate triples on any non-degenerate hypersurface of \(\mathbb{R}^{n+1}\). The paper under review studies the converse question: given a class of such conjugate triples on a simply connected manifold \(M^n\), when does there exist an immersion of \(M^n\) into \(\mathbb{R}^{n+ 1}\) inducing this class on \(M^n\). The answer is very elegant: if and only if one of the connections in the class is flat. The next question is to determine necessary and sufficient conditions for such classes to contain a flat connection. For this purpose, the author studies some invariant curvature tensors which vanish in case the class contains a flat connection.
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 53A20
Mathematics Subject Classification ID: 53A30
zbMATH DE Number: 785207
zbMATH Keywords: conformal structure
zbMATH Keywords: projective structure
zbMATH Keywords: affine hypersurface theory
zbMATH Keywords: relative normalization
reviewed by: Franki Dillen
MaRDI profile type: Publication
cites work: Conjugate connections and Radon's theorem in affine differential geometry
cites work: Q4867729
full work available at URL: https://doi.org/10.1007/bf03322280
OpenAlex ID: W2121259016
description: scientific article; zbMATH DE number 785207