\(\text{Spin}^q\), twistor and \(\text{Spin}^c\)
type: Item
instance of: scholarly article
title: \(\text{Spin}^q\), twistor and \(\text{Spin}^c\)
zbMATH Open document ID: 0907.53025
author: Masayoshi Nagase
published in: Communications in Mathematical Physics
publication date: 1999-02-18T00:00:00Z
review text: In an earlier paper [\textit{M. Nagase}, J. Math. Soc. Jap. 47, 93-119 (1995; Zbl 0863.57019)], the author introduced and studied the so-called \(\text{Spin}^q(n)\)-structures: the group \(\text{Spin}^q(n)\) is, by definition, \(\text{Spin}(n)\times_{{\mathbb Z}_2} \text{Sp}(1)\) and \(\text{Spin}^q(n)\)-structures for \(\text{SO}(n)\)-bundles are defined similarly to \(\text{Spin}(n)\)- or \(\text{Spin}^c(n)\)-structures. If \(P\to M\) is the principal \(\text{Spin}^q(n)\)-bundle of such a structure, the \({\mathbb C}P^1\)-bundle \(Z=P\times (\text{Spin}^q(n)/\text{Spin}^c(n))\) is called the (\(\text{Spin}^q(n)\)-style) twistor space associated to the given \(\text{Spin}^q(n)\)-structure. Any almost-quaternionic manifold \(M\) has a canonical \(\text{Spin}^q(n)\)-structure [op.cit.] whose twistor space coincides with the twistor space studied by \textit{S. Salamon} [Invent. Math. 67, 143-171 (1982; Zbl 0486.53048)] provided \(\dim M=8m+4\) and, in contrast to the Salamon twistor space, is trivial if \(\dim M=8m\) (since the canonical \(\text{Spin}^q(n)\)-structure is trivial in this case). In Section 1 of the present paper, the author shows that the twistor space \(Z\) associated to any \(\text{Spin}^q(n)\)-structure admits a natural \(\text{Spin}^c(n+2)\)-structure. Suppose the structure \(\text{SO}(n)\)-bundle of an oriented Riemannian manifold \((M,g)\) of dimension \(n\) admits a \(\text{Spin}^q(n)\)-structure. Any given connection of the principal \(SO(3)\)-bundle of this structure, together with the Levi-Civita connection of \(M\), induces a connection on the \(\text{Spin}^q(n)\)-bundle \(P\) of the structure and so on the associated bundle \(\pi:Z\to M\). The latter connection, as is usual in twistor theory, gives rise to a Riemannian metric \(g^Z= \pi^{*}g+ g^v\) on \(Z\) where \(g^v\) is the Fubini-Study metric on the fibre \({\mathbb C}P^1\) of \(Z\). In Section 2, the author considers the Levi-Civita connection of the metric \(g^Z\) and its relation wit…
Mathematics Subject Classification ID: 53C27
Mathematics Subject Classification ID: 53C28
Mathematics Subject Classification ID: 57R15
Mathematics Subject Classification ID: 53C07
zbMATH DE Number: 1121190
zbMATH Keywords: \(\text{Spin}^q\)-structures
zbMATH Keywords: twistors
reviewed by: Johann Davidov
MaRDI profile type: Publication
full work available at URL: s002200050192
OpenAlex ID: W1625719896
DOI: 10.1007/S002200050192
description: scientific article; zbMATH DE number 1121190