Abstract

Abstract Motivated by the recent interest in even-Clifford structures and in generalized complex and quaternionic geometries, we introduce the notion of generalized almost even-Clifford structure. We generalize the Arizmendi-Hadfield twistor space construction on even-Clifford manifolds to this setting and show that such a twistor space admits a generalized complex structure under certain conditions.

Highlights

  • Generalized complex, quaternionic and exceptional geometries have attracted attention in mathematics and physics for the last two decades [1, 7, 8, 14, 16, 20, 24, 26, 27, 29, 30, 34, 39, 43]

  • Motivated by the recent interest in even-Cli ord structures and in generalized complex and quaternionic geometries, we introduce the notion of generalized almost even-Cli ord structure

  • Pantilie [34] introduced the notion of generalized quaternionic manifold as well as the corresponding twistor space endowed with a connection-dependent generalized almost complex structure, whose integrability was discussed by Deschamps [17]

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Summary

Introduction

Generalized complex, quaternionic and exceptional geometries have attracted attention in mathematics and physics for the last two decades [1, 7, 8, 14, 16, 20, 24, 26, 27, 29, 30, 34, 39, 43]. Pantilie [34] introduced the notion of generalized quaternionic manifold as well as the corresponding twistor space endowed with a connection-dependent generalized almost complex structure, whose integrability was discussed by Deschamps [17]. There has been some interest in Riemannian manifolds admitting even-Cli ord structures [2,3,4,5, 11,12,13, 15, 18, 22, 23, 25, 28, 31,32,33, 35,36,37,38, 41] and, in particular, Arizmendi and Had eld [5] constructed a twistor space for even-Cli ord manifolds. We recall some concepts of generalized complex geometry [20] and Cli ord algebras [19]

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The twistor space construction
Xv and
Integrability of J
First notice
The second term is
The fourth term is
For a vertical form ξ
Such a section de nes a local almost complex structure by
IU t
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