  A similarity structure on a connected manifold M is a Riemannian metric on
its universal cover such that the fundamental group of M acts 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 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.
