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The characteristic group of locally conformally product structures

Brice Flamencourt

Abstract

A compact manifold $M$ together with a Riemannian metric $h$ on its universal cover $\tilde M$ for which $π_1(M)$ acts by similarities is called a similarity structure. In the case where $π_1(M) \not\subset \mathrm{Isom}(\tilde M, h)$ and $(\tilde M, h)$ 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 $\mathbb{R}^p$-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.

The characteristic group of locally conformally product structures

Abstract

A compact manifold together with a Riemannian metric on its universal cover for which acts by similarities is called a similarity structure. In the case where and 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 -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.
Paper Structure (13 sections, 19 theorems, 58 equations)

This paper contains 13 sections, 19 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General results
  4. Properties of some group actions on LCP manifolds
  5. Finite coverings of LCP manifolds
  6. Simple LCP manifolds
  7. Analysis of the LCP structure
  8. Construction of an LCP structure from admisible data
  9. Discussion on the hypotheses and further examples
  10. The orbifold hypothesis and the structure of the fundamental group
  11. The linear part of $\Omega$
  12. The non-constant translations
  13. Existence of the group $\Omega$.

Key Result

Theorem 2.1

Let $D$ be a closed, non-exact Weyl structure on a compact conformal manifold $(M, c)$. Assume moreover that $D$ is non-flat and has reducible holonomy. Then, there exists $q \ge 1$ and an irreducible incomplete Riemannian manifold $(N, g_N)$ such that the universal cover $\tilde{M}$ of $M$ endowed

Theorems & Definitions (38)

  • Theorem 2.1: Kourganoff
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • proof
  • Corollary 3.4
  • proof
  • Proposition 3.5
  • ...and 28 more