Lorentzian homogeneous structures with indecomposable holonomy

2024/04/26 by Greenwood, Steven, Leistner, Thomas
#53B30 #53C50 #Differential Geometry (math.DG) #FOS: Mathematics #Primary 53C30 #Secondary 53C29

paper · doi:10.48550/arxiv.2404.17470

Abstract

For a Lorentzian homogeneous space, we study how algebraic conditions on the isotropy group affect the geometry and curvature of the homogeneous space. More specifically, we prove that a Lorentzian locally homogeneous space is locally isometric to a plane wave if it admits an Ambrose--Singer connection with indecomposable, non-irreducible holonomy. This generalises several existing results that require a certain algebraic type of the torsion of the Ambrose--Singer connection and moreover is in analogy to the fact that a Lorentzian homogeneous space with irreducible isotropy has constant sectional curvature. In addition, we prove results about Lorentzian connections with parallel torsion and for 2-symmetric connections.

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