Holomorphic isometries of twistor spaces
type: Item
instance of: scholarly article
title: Holomorphic isometries of twistor spaces
zbMATH Open document ID: 1005.53032
DOI: 10.1016/S0393-0440(01)00067-5
author: Costantino Medori
author: Adriano Tomassini
published in: Journal of Geometry and Physics
publication date: 2002-06-23T00:00:00Z
review text: The twistor space of a smooth \(2n\)-dimensional Riemannian manifold \(M\) is the natural \(SO(2n)/\) \(U(n)\)-bundle \(Z(M)\) over \(M\). The Levi-Civita connection of \(M\) determines a connection and hence a natural Riemannian metric on \(Z(M)\) in such a way that the natural projection \(Z(M)\to M\) is a Riemannian submersion. The authors show that if the Ricci-tensor of \(M\) is parallel and non-positive then every isometry of \(Z(M)\) preserves the horizontal and vertical distributions. If in addition the Ricci tensor is negative then the Bochner formula shows that there are no nontrivial Killing fields on \(M\). As a consequence, in this case \(Z(M)\) does not admit a Killing field which is holomorphic with respect to the natural almost complex structure.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C56
zbMATH DE Number: 1758099
zbMATH Keywords: twistor spaces
zbMATH Keywords: Killing fields
zbMATH Keywords: holomorphic isometries
MaRDI profile type: Publication
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full work available at URL: s0393-0440(01)00067-5
OpenAlex ID: W2074807264
description: scientific article; zbMATH DE number 1758099