# On Weyl-reducible conformal manifolds and lcK structures

## TL;DR

This paper explores the structure of Weyl-reducible conformal manifolds, showing restrictions on their universal covers and characterizing cases related to Anosov diffeomorphisms, thus advancing understanding of their geometric and dynamical properties.

## Contribution

It extends Kourganoff's results by establishing that the parameter q must be 1 or 2 in certain Weyl-reducible conformal manifolds, refining the classification of these structures.

## Key findings

- Universal cover splits as R^q times an incomplete manifold N
- If dim(N)=2, then q=1 or 2
- Provides constraints on the structure of Weyl-reducible conformal manifolds

## Abstract

A recent result of M. Kourganoff states that if $D$ is a closed, reducible, non-flat, Weyl connection on a compact conformal manifold $M$, then the universal covering of $M$, endowed with the metric whose Levi-Civita covariant derivative is the pull-back of $D$, is isometric to $\mathbb{R}^q\times N$ for some irreducible, incomplete Riemannian manifold $N$. Moreover, he characterized the case where the dimension of $N$ is $2$ by showing that $M$ is then a mapping torus of some Anosov diffeomorphism of $T^{q+1}$. We show that in this case one necessarily has $q=1$ or $q=2$.