Holonomy groups of compact flat solvmanifolds

Abstract

In this article we study the holonomy group of flat solvmanifolds. It is known that the holonomy group of a flat solvmanifold is abelian; we give an elementary proof of this fact and moreover we prove that any finite abelian group is the holonomy group of a flat solvmanifold. Furthermore, we show that the minimal dimension of a flat solvmanifold with holonomy group Z n coincides with the minimal dimension of a compact flat manifold with holonomy group Z n . Finally, we give the possible holonomy groups of flat solvmanifolds in dimensions 3, 4, 5 and 6; exhibiting in the latter case a general construction to show examples of non cyclic holonomy groups.

Classification of 6-dimensional splittable flat solvmanifolds

“…Remark 6.2. In [24], we proved that an almost abelian flat solvmanifold has cyclic holonomy group. In view of Corollary 6.1 one can conjecture that the converse also holds, i.e., if a splittable flat solvmanifold has cyclic holonomy group then the solvmanifold is diffeomorphic to an almost abelian one.…”

“…In [24] we described the structure of an almost abelian flat Lie algebra. Then we can write an almost abelian flat Lie algebra g as g = Rx ⋉ adx R s+2n with s = dim z(g), 2n = dim[g, g] and in some basis B of R s+2n we have…”

“…In a previous paper ( [24]), we studied flat solvmanifolds starting from the point of view of the well known Milnor's characterization of solvable Lie groups which admit a flat left-invariant metric. We combined this with Bieberbach's classical theory to prove some properties of the holonomy group of a flat solvmanifold.…”

“…The main goal of this article is to continue the study initiated in [24] and, in particular, to provide the classification of splittable flat solvmanifolds in dimension 6. In §3 we outline some facts about splittable solvmanifolds.…”

“…[24, Theorem 3.3] Let g = b ⊕ z(g) ⊕ [g, g]be a flat Lie algebra. Then g is almost abelian if and only if dim b = 1.…”