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Holonomy groups of compact flat solvmanifolds

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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 \({\mathbb {Z}}_n\) coincides with the minimal dimension of a compact flat manifold with holonomy group \({\mathbb {Z}}_n\). 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.

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Notes

  1. An affine equivalence between two Riemannian manifolds M and N equipped with connections \(\nabla ^M\) and \(\nabla ^N\) respectively, is a diffeomorphism \(F:M\rightarrow N\) such that \(\nabla _X^M Y=\nabla ^N_{dFX}(dF Y)\) for all \(X,Y\in {\mathfrak {X}}(M)\).

  2. A Lie algebra \({\mathfrak {g}}\) is said to be unimodular if \({\text {tr}}{\text {ad}}_X=0\) for all \(X\in {\mathfrak {g}}\).

  3. We will denote by \(A=A_1\oplus A_2\) the block diagonal matrix in \({\mathbb {R}}^4\) given by \(A=\begin{pmatrix}A_1&{}\\ {} &{}\quad A_2\end{pmatrix}\).

  4. The values for \(at_0\) and \(bt_0\) are interchangeable.

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  1. FaMAF, Universidad Nacional de Córdoba, Ciudad Universitaria, X5000HUA, Córdoba, Argentina

    A. Tolcachier

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Tolcachier, A. Holonomy groups of compact flat solvmanifolds. Geom Dedicata 209, 95–117 (2020). https://doi.org/10.1007/s10711-020-00524-8

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