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The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1075–1108 | Cite as

Holonomy Classification of Lorentz-Kähler Manifolds

  • Anton S. GalaevEmail author
Article

Abstract

The classification problem for holonomy of pseudo-Riemannian manifolds is actual and open. In the present paper, holonomy algebras of Lorentz-Kähler manifolds are classified. A simple construction of a metric for each holonomy algebra is given. Complex Walker coordinates are introduced and described using the potential. Complex pp-waves are characterized in terms of the curvature, holonomy, and the potential. Classification of Lorentz-Kähler symmetric spaces is reviewed.

Keywords

Lorentz-Kähler manifold Holonomy group Complex Walker coordinates Complex pp-wave Symmetric space Space of oriented lines 

Mathematics Subject Classification

53C29 53C25 53C35 53C50 53C55 

Notes

Acknowledgements

The author is grateful to Helga Baum and Dmitri V. Alekseevsky for useful discussions and suggestions. This work was supported by the Specific Research Project of Faculty of Science, University of Hradec Králové [No. 2101, 2017] and by the grant no. 18-00496S of the Czech Science Foundation.

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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Faculty of ScienceUniversity of Hradec KrálovéHradec KrálovéCzech Republic

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