Affine manifold with measure preserving projective holonomy group
type: Item
instance of: scholarly article
zbMATH Open document ID: 1009.53037
author: Kyeongsu Park
publication date: 2003-05-06T00:00:00Z
Mathematics Subject Classification ID: 53C29
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 57N16
zbMATH DE Number: 1594810
zbMATH Keywords: affine manifold
zbMATH Keywords: affine holonomy
zbMATH Keywords: radiant manifold
zbMATH Keywords: Euler characteristic
MaRDI profile type: Publication
title: Affine manifold with measure preserving projective holonomy group
published in: Bulletin of the Korean Mathematical Society
review text: The term radiance was introduced by \textit{D. Fried}, \textit{W. Goldman} and \textit{M. W. Hirsch} [Comment. Math. Helv. 56, No. 4, 487-523 (1981; Zbl 0516.57014)] as an affine manifold whose affine holonomy has a fixed point. In this paper the author considers a compact affinely flat manifold \(M\) covered by a radiant manifold \(M\) and proves that the Euler characteristic \(\chi(M)\) is zero, in case the projective holonomy group of \(M\) leaves a probability Borel measure invariant. Otherwise the affine tangent bundle of \(\overline M\) contains a conformally flat vector subbundle, that is a linear subbundle with holonomy in the group of similarity transformations. Also in this case \(\chi(M)=0\). The paper gives another way of proving the theorem of \textit{M. W. Hirsch} and \textit{W. P. Thurston} [Ann. Math. (2) 101, 369-390 (1975; Zbl 0321.57015)] related particularly to amenability, a notion due to \textit{W. M. Goldman} and \textit{M. W. Hirsch} [Proc. Am. Math. Soc. 82, No. 3, 491-494 (1981; Zbl 0474.55014)].
reviewed by: Demetra Demetropoulou-Psomopoulou
description: scientific article