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.
It is well-known that the product of two Sasakian manifolds carries a 2-parameter family of Hermitian structures (J a,b , g a,b ). We show in this article that the complex structure J a,b is harmonic with respect to g a,b , i.e. it is a critical point of the energy functional. Furthermore, we also determine when these Hermitian structures are locally conformally Kähler, balanced, strong Kähler with torsion, Gauduchon or k-Gauduchon (k ≥ 2). Finally, we provide an expression for the Bismut connection associated to (J a,b , g a,b ) in terms of the characteristic connections on each Sasakian factor.
In this article we study the relation between flat solvmanifolds and G 2geometry. First, we give a classification of 7-dimensional flat splittable solvmanifolds using the classification of finite subgroups of GL(n, Z) for n = 5 and n = 6. Then, we look for closed, coclosed and divergence-free G 2 -structures compatible with the flat metric on them. In particular, we provide explicit examples of compact flat manifolds with a torsion-free G 2 -structure whose finite holonomy is cyclic and contained in G 2 , and examples of compact flat manifolds admitting a divergence-free G 2 -structure.
A flat solvmanifold is a compact quotient Γ\G where G is a simply-connected solvable Lie group endowed with a flat left invariant metric and Γ is a lattice of G. Any such Lie group can be written as G = R k ⋉ φ R m with R m the nilradical. In this article we focus on 6-dimensional splittable flat solvmanifolds, which are obtained quotienting G by a lattice Γ that can be decomposed as Γ = Γ 1 ⋉ φ Γ 2 , where Γ 1 and Γ 2 are lattices of R k and R m , respectively. We obtain their classification by analyzing the conjugacy classes of integer matrices of finite order in dimensions 4 and 5.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.