The characteristic group of locally conformally product structures

2024/01/16 by Brice Flamencourt, Flamencourt, Brice · 2 citations
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2401.08372

Abstract

A compact manifold M together with a Riemannian metric h on its universal cover M for which π₁(M) acts by similarities is called a similarity structure. In the case where π₁(M) \not⊂ Isom( M, h) and ( M, h) is reducible but not flat, this is a Locally Conformally Product (LCP) structure. The so-called characteristic group of these manifolds, which is a connected abelian Lie group, is the key to understand how they are built. We focus in this paper on the case where this group is simply connected, and give a description of the corresponding LCP structures. It appears that they are quotients of trivial ℝ^p-principal bundle over simply-connected manifolds by certain discrete subgroups of automorphisms. We prove that, conversely, it is always possible to endow such quotients with an LCP structure.

Cited by

• The characteristic group of Lie LCP manifolds
• Conformal product structures on compact manifolds with constant sectional curvature

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