Holonomy groups of compact flat solvmanifolds

2019/07/03 by Tolcachier, Alejandro
#20H15 #22E25 #22E40 #53C29 #Differential Geometry (math.DG) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1907.02021

Abstract

This article is concerned with the study of 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.

Related

• G₂-structures on flat solvmanifolds
• Flat manifolds with holonomy representation of quaternionic type
• Holonomy Groups in Riemannian Geometry
• Seven dimensional flat manifolds with cyclic holonomy
• Projectively flat Finsler manifolds with infinite dimensional holonomy