The only Kähler manifold with degree of mobility at least 3 is \((\mathbb CP(n), g_{\text{Fubini-Study}})\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1248.53054
author: A. Fedorova
author: S. Rosemann
author: V. Kiosak
author: Vladimir S. Matveev
publication date: 2012-07-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1009.5530
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C17
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 32J27
Mathematics Subject Classification ID: 53A20
zbMATH DE Number: 6062403
zbMATH Keywords: \(h\)-planar
zbMATH Keywords: \(h\)-projective equivalent
zbMATH Keywords: degeree of mobility of a Kähler metric
MaRDI profile type: Publication
OpenAlex ID: W3104373656
arXiv ID: 1009.5530
DOI: 10.1112/PLMS/PDR053
title: The only Kähler manifold with degree of mobility at least 3 is \((\mathbb CP(n), g_{\text{Fubini-Study}})\)
published in: Proceedings of the London Mathematical Society. Third Series
review text: Let \(g\) be a (possible non positive definite) Kähler metric on a complex manifold \(M\) with complex structure \(J.\) A curve \(\gamma:I\rightarrow M\) is called \(h\)-planar is there are functions \(\alpha(t),\beta(t)\) such that NEWLINE\[NEWLINE\nabla _{\gamma'(t)}{\gamma'(t)}=\alpha(t)\gamma'(t)+\beta(t)J(\gamma'(t)).NEWLINE\]NEWLINE Two Kähler metrices \(g\) and \(\bar{g}\) with the same complex structure are called \(h\)-projectively equivalent, if each \(h\)-planar curve of \(g\) is an \(h\)-planar curve of \(\bar{g}\) and vice versa. The degree of mobility of a Kähler metric \(g\) measures the dimension of the space of Kähler metrics which are \(h\)-projectively equivalent to \(g\).NEWLINENEWLINEThe paper under review proves that a metric on a closed connected manifold can not have the degree of mobility at least 3 unless it is essentially the Fubini-Study metric, or the h-projective equivalence is actually the affine equivalence. As main application it proves an important special case of the classical conjecture attributed to Obata and Yano, stating that a closed manifold admitting an essential group of \(h\)-projective transformations is (\({\mathbb C}{\mathbb P}(n), g_{\text{Fubini.Study}})\) (up to multiplication of the metric by a constant). An additional result is the generalization of a certain result of Tanno 1978 for the pseudo-Riemannian situation.
reviewed by: Qilin Yang
description: scientific article; zbMATH DE number 6062403