Holonomy groups of compact flat solvmanifolds
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Holonomy groups of compact flat solvmanifolds A. Tolcachier1 Received: 22 July 2019 / Accepted: 10 March 2020 © Springer Nature B.V. 2020
Abstract In this article we study the holonomy groups 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 Zn coincides with the minimal dimension of a compact flat manifold with holonomy group Zn . 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. Keywords Bieberbach group · Holonomy · Solvable Lie group · Solvmanifold · Lattice Mathematics Subject Classification (2010) 20H15 · 22E25 · 22E40 · 53C29
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Bieberbach groups and compact flat manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Lie groups and flat solvmanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Holonomy of flat solvmanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Minimal dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Low dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Dimension 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Dimension 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Dimension 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Dimension 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A. Tolcachier [email protected] FaMAF, Universidad Nacional de Córdoba, Ciudad Universitaria, X5000HUA Córdoba, Argentina
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Geometriae Dedicata
1 Introduction A solvmanifold is defined as a compact homogeneous space G\Γ of a simply connected solvable Lie group G by a discrete subgroup Γ . Solvmanifolds generalize the well known nilmanifolds which are defined similarly when G is nilpotent. Both nilmanifolds and solvmanifolds have provided a large number of examples and counterexamples in differential geometry. For instance, the first example of a symplectic manifold without Kähler structure, the so-called “Kodaira–Thurston manifold”, is a four dimensional nilmanifold [29]. However, many important global properties of nilmanifolds cannot be generalized to solv
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