Transitive conformal holonomy groups

2011/07/04 by Jesse Alt, Alt, Jesse
Mathematics · #Advanced Algebra and Geometry #Geometry and complex manifolds #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.1107.0617

Abstract

For (M,[g]) a conformal manifold of signature (p,q) and dimension at least three, the conformal holonomy group Hol(M,[g]) ⊂ O(p+1,q+1) is an invariant induced by the canonical Cartan geometry of (M,[g]). We give a description of all possible connected conformal holonomy groups which act transitively on the Möbius sphere S^p,q, the homogeneous model space for conformal structures of signature (p,q). The main part of this description is a list of all such groups which also act irreducibly on \R^p+1,q+1. For the rest, we show that they must be compact and act decomposably on \R^p+1,q+1, in particular, by known facts about conformal holonomy the conformal class [g] must contain a metric which is locally isometric to a so-called special Einstein product.

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