Locally conformally Berwald manifolds and compact quotients of reducible manifolds by homotheties

TL;DR

This paper characterizes certain locally conformally Berwald metrics on closed manifolds by linking them to incomplete, reducible holonomy Riemannian manifolds and establishes a splitting theorem for analytic cases.

Contribution

It provides a new characterization of non-globally conformally Berwald metrics via holonomy and quotient properties, and proves a splitting theorem for analytic cases.

Findings

Characterization of non-globally conformally Berwald metrics via holonomy groups.

Equivalence between such metrics and quotients of incomplete, reducible holonomy manifolds.

A de Rham type splitting theorem for analytic manifolds.

Abstract

We study locally conformally Berwald metrics on closed manifolds which are not globally conformally Berwald. We prove that the characterization of such metrics is equivalent to characterizing incomplete, simply-connected, Riemannian manifolds with reducible holonomy group whose quotient by a group of homotheties is closed. We further prove a de Rham type splitting theorem which states that if such a manifold is analytic, it is isometric to the Riemannian product of a Euclidean space and an incomplete manifold.