Similarity structures and de Rham decomposition

Abstract

A similarity structure on a connected manifold M is a Riemannian metric on its universal coverM such that the fundamental group of M acts onM by similarities. If the manifold M is compact, we show that the universal cover admits a de Rham decomposition with at most two factors, one of which is Euclidean. Very recently, after Belgun and Moroianu conjectured that the number of factors was at most one, Matveev and Nikolayevsky found an example with two factors. When the non-flat factor has dimension 2, we give a complete classification of the examples with two factors. In greater dimensions, we make the first steps towards such a classification by showing that M is a fibration (with singularities) by flat Riemannian manifolds. Up to a finite covering of M , we may assume that these manifolds are flat tori. We also prove a version of the de Rham decomposition theorem for the universal covers of manifolds endowed with locally metric connections. During the proof, we define a notion of transverse (not necessarily flat) similarity structure on foliations, and show that foliations endowed with such a structure are either transversally flat or transversally Riemannian. None of these results assumes analyticity.

Locally conformally product structures

“…More recently, Kourganoff extended this result to the smooth setting [4,Theorem 1.5]. More precisely, he proved that if a closed, non-exact Weyl structure D on a compact conformal manifold (M, c) is non-flat and has reducible holonomy, then the Riemannian manifold ( M , h D ) is isometric to the Riemannian product R q × (N, g N ) where R q (the flat part) is an Euclidean space and (N, g N ) (the non-flat part) is an irreducible, non-complete manifold.…”

“…We also remind some basics about algebraic number fields, which will be needed in the sequel. Indeed, it turns out that the study of LCP manifolds is closely related to number theory, a fact that we can already notice from the previous examples, which involve matrices in GL n (Z) [4] and algebraic number fields [11]. The structure theorem for LCP manifold proved by Kourganoff [4, Theorem 1.9], is also restated.…”

“…There are up to now only few examples of LCP manifolds. As mentioned before, the first one was given in [7], and generalized in [4,Example 1.6] (we outline the construction in Example 2.7 below). This example is very restrictive because it only provides LCP manifolds of dimension 3 or 4, with a flat part of dimension 1 or 2 [6].…”

“…This example is very restrictive because it only provides LCP manifolds of dimension 3 or 4, with a flat part of dimension 1 or 2 [6]. Nevertheless, they are the only examples when the non-flat part is of dimension 2 [4,Theorem 1.8]. The other class of example comes from the theory of locally conformally Kähler (or LCK) manifolds.…”

On Weyl-reducible conformal manifolds and lcK structures

“…In [6,Example 1.6] (see also [7]), examples of closed reducible Weyl structures on compact manifolds are constructed using a linear map A ∈ SL q+1 (Z), such that:…”