Holonomy groups of compact flat solvmanifolds

TL;DR

This paper investigates the holonomy groups of flat solvmanifolds, proving they are abelian, can realize any finite abelian group, and classifying possible groups in low dimensions.

Contribution

It provides an elementary proof that holonomy groups are abelian and constructs examples for all finite abelian groups as holonomy groups.

Findings

Holonomy groups of flat solvmanifolds are abelian.

Any finite abelian group can be realized as a holonomy group.

Classification of possible holonomy groups in dimensions 3 to 6.

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 $\mathbb{Z}_n$ 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.

Equations

$$\mathfrak{g}=\mathfrak{z}(\mathfrak{g})\oplus \mathfrak{b}\oplus [\mathfrak{g},\mathfrak{g}]$$

$$[{\mathrm{ad}}_X{]}_{\mathcal{B}}=\left(\begin{array}{ccccc}0& a_1& & & \\ -a_1& 0& & & \\ & & \ddots & & \\ & & & 0& a_m\\ & & & -a_m& 0\end{array}\right),\text{with}{a}_1,\dots ,a_m\in \mathbb{R},m=\frac{\dim [\mathfrak{g},\mathfrak{g}]}{2}.$$

$$⟨u,v{⟩}_{\pi (g)}=⟨(d\pi {)}_g^{-1}u,(d\pi {)}_g^{-1}v{⟩}_g.$$

$$\Gamma =t_0\mathbb{Z}{⋉}_{\phi }P{\mathbb{Z}}^m,$$

$$\varphi ([\gamma g])=[\psi (\gamma g)]=[{\tau }_{\gamma }(\psi (g))]=[\psi (g)]=\varphi ([g]).$$

$$\mathrm{Hol}(\Gamma \backslash G)\cong \Gamma /\Lambda .$$

Full text

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 Bieberbach groups and compact flat manifolds

4. 2.2 Lie groups and flat solvmanifolds

5. 3 Holonomy of flat solvmanifolds

1. Introduction

A solvmanifold is defined as a compact homogeneous space $G\backslash\varGamma$ of a simply connected solvable Lie group $G$ by a discrete subgroup $\varGamma$. 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 [Th]. However, many important global properties of nilmanifolds cannot be generalized to solvmanifolds, and for this reason these manifolds are currently widely studied. One of the most important features of nilmanifolds is a result proved by Nomizu, which says briefly that the cohomology of the nilmanifolds can be computed using invariant differential forms, since it is isomorphic to the cohomology of its Lie algebra [No54]. Unfortunately this is not true in general for solvmanifolds, but only in particular cases. For example, when $G$ is completely solvable [Hat] or when the “Mostow condition” holds [Mo61].

Solvmanifolds have had several applications in complex geometry. For instance let us mention the well known Oeljeklaus-Toma manifolds. In [OT], for any integers $s>0$ and $t>0$, Oeljeklaus and Toma constructed compact complex manifolds by using number theory and they showed that for $t=1$ these manifolds admit locally conformally Kähler metrics; they provide a counterexample to a conjecture made by Vaisman. Kasuya proved in [K] that OT-manifolds can be represented as solvmanifolds and using this description he proved that they do not admit any Vaisman metric.

Solvmanifolds have also had applications in theoretical physics, for instance in [FMOU], solutions of the supersymmetric equations of type II A are exhibited on six-dimensional solvmanifolds admitting an invariant symplectic half-flat structure. In [OUV] the authors provide an example of a six-dimensional complex solvmanifold which admits many invariant solutions to the Strominger system with respect to certain family of Hermitian connections in the Green-Schwarz anomaly cancelation condition. Moreover, some of these solutions satisfy in addition the heterotic equations of motion.

We are interested in flat solvmanifolds, i.e, solvmanifolds admitting a flat Riemannian metric induced by a flat left invariant metric on the corresponding Lie group. Milnor [Mi] gave a characterization of Lie groups which admit a flat left invariant metric and he showed that they are all solvable of a very restricted form, proving that its Lie algebra decomposes orthogonally as an abelian subalgebra and an abelian ideal, where the action of the subalgebra on the ideal is by skew-adjoint endomorphisms. Moreover, such a Lie group $G$ is of type (I), that is the eigenvalues of $\operatorname{ad}_{x}$ are pure imaginary for all $x\in\operatorname{Lie}(G)$, and in particular $G$ is unimodular. Some simply connected Lie groups of this class admit lattices (i.e. cocompact discrete subgroups), therefore the corresponding solvmanifolds admit a flat Riemannian metric induced by the flat left invariant metric on the Lie group.

Flat solvmanifolds are a special class of compact flat manifolds. A classical theorem asserts that a complete Riemannian manifold is flat if and only if its universal covering is isometric to Euclidean space. In particular, the manifold is isometric to a quotient of the form $\mathbb{R}^{n}/\Gamma$ for some discrete subgroup $\Gamma$ of $\operatorname{Iso}(\mathbb{R}^{n})$, and its fundamental group is isomorphic to $\Gamma$. In the compact case, such subgroups are well described by the well known Bieberbach’s theorems ([Bi1, Bi2, Bi3]) and consequently they are called “Bieberbach groups”. Particularly, a Bieberbach group $\Gamma$ admits a unique maximal normal abelian subgroup $\Lambda$ of finite index. Furthermore, the finite group $\Gamma/\Lambda$ can be identified with the Riemannian holonomy group of the compact flat manifold. A famous result of Auslander and Kuranishi [AusK, Theorem 3] establishes that every finite group is the holonomy group of a compact flat manifold.

The fundamental group of a solvmanifold $\varGamma\backslash G$ is isomorphic to the lattice $\varGamma$ and when the solvmanifold is flat it is isomorphic to a Bieberbach group. In [Aus], fundamental groups of flat solvmanifolds were characterized.

The purpose of this article is to study the holonomy group of flat solvmanifolds. It was proved in [Aus] that these finite holonomy groups are abelian. In §3, by using Milnor’s result mentioned above, we provide an elementary proof of this fact (Theorem LABEL:holabeliana). The main result in this section is the proof of the converse. In order to do this, we focus on a special class of flat solvmanifolds which are obtained as quotients of almost abelian Lie groups, since for these Lie groups there exists a criterion to determine the existence of lattices (see [B]). We show that the holonomy group of a flat almost abelian solvmanifold is cyclic (Theorem LABEL:holonomia_lcm), and furthermore for $n=p^{k}$ and $n=2p^{k}$, with $p$ a prime and $k\in\mathbb{N}$, we give an explicit construction of an almost abelian Lie group and a lattice such that the quotient solvmanifold has holonomy group $\mathbb{Z}_{n}$ (Theorem LABEL:holpotprimos). Combining these results we prove the converse, namely, that every finite abelian group is the holonomy group of a flat solvmanifold (Theorem LABEL:holabelianofinito).

Due to the result of Auslander and Kuranishi, it is interesting to know, given a finite group $G$, the minimal dimension of a compact flat manifold with holonomy group $G$. The minimal dimension was determined for several groups, for example for $\mathbb{Z}_{p}$ with $p$ prime in [Ch], and more generally for cyclic groups in [Hi]. Concerning this, we prove in §4 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 $\mathbb{Z}_{n}$, for $n\geq 3$ (Theorem LABEL:mindim).

The last section of the article (§5) is devoted to studying holonomy groups of low-dimensional flat solvmanifolds. By using a refined version of Milnor’s result proved by Barberis, Dotti and Fino in [BDF], we give the possible holonomy groups of flat solvmanifolds in dimensions 3, 4, 5 and 6. Up to dimension 5 all flat solvmanifolds are quotients of almost abelian Lie groups, therefore their holonomy groups are cyclic. On the other hand, in dimension 6 we exhibit examples with non-cyclic abelian holonomy groups as particular cases of a general construction.

The results of this article are from part of my bachelor thesis under the direction of Adrian Andrada. I would like to thank Roberto Miatello and Juan Pablo Rossetti for their useful suggestions and comments.

2. Preliminaries

In this short section we outline the basic theory of the topics studied in this article, namely compact flat manifolds and solvmanifolds equipped with invariant flat metrics.

2.1. Bieberbach groups and compact flat manifolds

It is well known that the group of isometries of $\mathbb{R}^{n}$ is $\operatorname{Iso}(\mathbb{R}^{n})\cong{\sf O}(n)\ltimes_{\varphi}\mathbb{R}^{n}$, where $\varphi(A)(v)=Av$. The pair $(A,v)$ is identified with the isometry $f$ of $\mathbb{R}^{n}$ given by $f(x)=Ax+v$. This group is called the group of rigid transformations of $\mathbb{R}^{n}$. Equipped with the product topology of ${\sf O}(n)$ and $\mathbb{R}^{n}$, $\operatorname{Iso}(\mathbb{R}^{n})$ is a topological group. The affine group $\mathcal{A}_{n}$ is defined by $\mathcal{A}_{n}={\sf GL}(n,\mathbb{R})\ltimes_{\varphi}\mathbb{R}^{n}$. An element $(A,v)\in\operatorname{Iso}(\mathbb{R}^{n})$ is called a pure translation if $A=\operatorname{Id}$. If $\Gamma$ is a subgroup of $\operatorname{Iso}(\mathbb{R}^{n})$, the subgroup $\Gamma\cap\mathbb{R}^{n}$ of pure translations of $\Gamma$ will be usually denoted by $\Lambda$, and it is a torsionfree abelian normal subgroup of $\Gamma$.

Definition 2.1**.**

A subgroup $\Gamma$ of $\operatorname{Iso}(\mathbb{R}^{n})$ is called co-compact if $\mathbb{R}^{n}/\Gamma$ is compact. A co-compact and discrete subgroup $\Gamma$ is called a crystallographic group. If $\Gamma$ is also torsionfree then $\Gamma$ is called a Bieberbach group.

For a discrete subgroup $\Gamma\subset\operatorname{Iso}(\mathbb{R}^{n})$, an equivalent condition to be torsionfree is that the action of $\Gamma$ on $\mathbb{R}^{n}$ be free. A nice class of group actions are the properly discontinuous actions. In general, we say that an action of a subgroup $G$ of the group of homeomorphisms of a topological space $X$ is properly discontinuous if for all $x\in X$ there exists an open neighborhood $U$ of $x$ such that $gU\cap U=\emptyset$ for all $g\neq e$. It is known that if a discrete subgroup $\Gamma$ of $\operatorname{Iso}(\mathbb{R}^{n})$ acts freely on $\mathbb{R}^{n}$, then the action is properly discontinuous and $\mathbb{R}^{n}/\Gamma$ admits a structure of a differentiable manifold. As a corollary, we have the next important theorem about Bieberbach groups.

Theorem 2.2**.**

Let $\Gamma$ be a Bieberbach subgroup of $\operatorname{Iso}(\mathbb{R}^{n})$ and $\pi:\mathbb{R}^{n}\to\mathbb{R}^{n}/\Gamma$ the canonical projection. Then $\Gamma$ acts on $\mathbb{R}^{n}$ in a properly discontinuous manner, hence $\pi$ is a covering map and $\pi_{1}(\mathbb{R}^{n}/\Gamma)\cong\Gamma$. Furthermore $\mathbb{R}^{n}/\Gamma$ admits a structure of a $n$-dimensional differentiable manifold such that $\pi$ is a local diffeomorphism.

Bieberbach groups, and more generally crystallographic groups, are well described by three classic theorems known as “Bieberbach’s theorems”. A proof of them can be found in [Ch].

Theorem 2.3** (Bieberbach’s First Theorem).**

Let $\Gamma$ be a crystallographic subgroup of $\operatorname{Iso}(\mathbb{R}^{n})$ and $\Lambda$ the subgroup of pure translations of $\Gamma$. Then $\Lambda$ is a normal free abelian subgroup of rank $n$ such that $\Gamma/\Lambda$ is a finite group. Furthermore, $\Lambda$ is maximal as a normal abelian subgroup, in the sense that it contains every other normal abelian subgroup of $\Gamma$.

The group $\Gamma/\Lambda$ is known as the point group of $\Gamma$ or the holonomy group of $\Gamma$, in the case that $\Gamma$ is a Bieberbach subgroup of $\operatorname{Iso}(\mathbb{R}^{n})$. The point group of $\Gamma$ can be identified with $r(\Gamma)\subset{\sf O}(n)$, which is a finite group, where $r$ is the projection homomorphism given by $r(A,v)=A.$ Indeed, the kernel of $r:\Gamma\to{\sf O}(n)$ is $\Lambda$ and thus $\Gamma/\Lambda\cong r(\Gamma)$.

Theorem 2.4** (Bieberbach’s Second Theorem).**

Let $\Gamma_{1}$ and $\Gamma_{2}$ be crystallographic subgroups of $\operatorname{Iso}(\mathbb{R}^{n})$ and let $f:\Gamma_{1}\to\Gamma_{2}$ a group isomorphism. Then there exists $\alpha\in\mathcal{A}_{n}$ such that $f(\beta)=\alpha\beta\alpha^{-1}$ for all $\beta\in\Gamma_{1}$. In other words, every isomorphism of crystallographic groups can be realized by an affine change of coordinates.

Theorem 2.5** (Bieberbach’s Third Theorem).**

For each $n\in\mathbb{N}$, there are only finitely many isomorphism classes of crystallographic subgroups of $\operatorname{Iso}(\mathbb{R}^{n})$.

Bieberbach’s theorems are fundamental tools to study compact flat manifolds. Indeed, a classical theorem asserts that a Riemannian manifold $M$ is flat if and only if $M$ is isometric to $\mathbb{R}^{n}/\Gamma$ where $\Gamma$ is a subgroup of $\operatorname{Iso}(\mathbb{R}^{n})$ that acts on $\mathbb{R}^{n}$ in a properly discontinuous manner, that is to say, $\Gamma$ is a Bieberbach subgroup of $\operatorname{Iso}(\mathbb{R}^{n})$. Furthermore, the holonomy group of the Riemannian manifold $\mathbb{R}^{n}/\Gamma$ can be identified with what is called the holonomy group of the Bieberbach group, that is $\operatorname{Hol}(\mathbb{R}^{n}/\Gamma)\cong r(\Gamma)$, see for instance [Ch, Example 3.1].

We can reinterpret Bieberbach’s theorems in the context of compact flat manifolds.

Theorem 2.6** (Bieberbach First).**

Let $M$ a compact flat manifold. Then $M$ admits a Riemannian covering by a flat torus and the covering is a local isometry. Furthermore, the holonomy group $\operatorname{Hol}(M)$ is finite.

Theorem 2.7** (Bieberbach Second).**

Let $M$ and $N$ be compact flat manifolds with isomorphic fundamental groups. Then there exists an affine equivalence111An affine equivalence between two Riemannian manifolds $M$ and $N$ equipped with connections $\nabla^{M}$ and $\nabla^{N}$ respectively, is a diffeomorphism $F:M\to N$ such that $\nabla_{X}^{M}Y=\nabla^{N}_{dFX}(dFY)$ for all $X,Y\in\mathfrak{X}(M)$. between $M$ and $N$ equipped with the corresponding Levi-Civita connections.

Theorem 2.8** (Bieberbach Third).**

For each $n\in\mathbb{N}$, there are only finitely many classes of affine equivalence of compact flat manifolds of dimension $n$.

2.2. Lie groups and flat solvmanifolds

In the context of Lie groups, the most interesting metrics are the left invariant ones, which satisfy that the left translations are isometries. When $G$ is an abelian Lie group, it is clear that every left invariant metric is flat. An interesting fact proved by Milnor is that some left invariant metrics may be flat, even though $G$ is not abelian. Indeed, he described the structure of a Lie group admitting one of these flat metrics (see [Mi, Theorem 1.5]). The precise criterion is stated as follows.

Theorem 2.9**.**

A left invariant metric on a Lie group $G$ is flat if and only if the associated Lie algebra $\mathfrak{g}$ splits as an orthogonal direct sum $\mathfrak{g}=\mathfrak{b}\oplus\mathfrak{u}$, where $\mathfrak{b}$ is an abelian subalgebra, $\mathfrak{u}$ is an abelian ideal and the linear transformation $\operatorname{ad}_{b}$ is skew-adjoint for every $b\in\mathfrak{b}$.

We will call $(G,\langle\cdot,\cdot\rangle)$ a flat Lie group if $\langle\cdot,\cdot\rangle$ is a flat left invariant metric on the Lie group $G$ and $(\mathfrak{g},\langle\cdot,\cdot\rangle_{e})$ will be called a flat Lie algebra.

Using Milnor’s characterization of flat Lie algebras, Barberis, Dotti and Fino decompose even more a flat Lie algebra [BDF, Proposition 2.1].

Theorem 2.10**.**

Let $(\mathfrak{g},\langle\cdot,\cdot\rangle_{e})$ be a flat Lie algebra. Then $\mathfrak{g}$ splits as an orthogonal direct sum,

[TABLE]

where $\mathfrak{b}$ is an abelian subalgebra, $[\mathfrak{g},\mathfrak{g}]$ is abelian and the following conditions are satisfied:

1. (1)

$\operatorname{ad}:\mathfrak{b}\to\mathfrak{so}[\mathfrak{g},\mathfrak{g}]$* is injective, $\dim[\mathfrak{g},\mathfrak{g}]$ is even and $\dim\mathfrak{b}\leq\frac{\dim[\mathfrak{g},\mathfrak{g}]}{2}$;* 2. (2)

$\nabla_{X}=\operatorname{ad}_{X}$* for all $X\in\mathfrak{b}$ and $\nabla_{X}=0$ for all $X\in\mathfrak{z}(\mathfrak{g})\oplus[\mathfrak{g},\mathfrak{g}]$.*

As a consequence, since $\{\operatorname{ad}_{X}\mid X\in\mathfrak{b}\}$ is an abelian subalgebra of $\mathfrak{so}[\mathfrak{g},\mathfrak{g}]$, it is included in a maximal abelian subalgebra and as all of these are conjugated, there exists an orthonormal basis $\mathcal{B}$ of $[\mathfrak{g},\mathfrak{g}]$ such that for $X\in\mathfrak{b}$,

[TABLE]

Note that a flat Lie algebra $(\mathfrak{g},\langle\cdot,\cdot\rangle_{e})$ is $2$-step solvable, since $[\mathfrak{g},\mathfrak{g}]$ is abelian; and unimodular222A Lie algebra $\mathfrak{g}$ is said to be unimodular if $\operatorname{tr}\operatorname{ad}_{X}=0$ for all $X\in\mathfrak{g}$., since $\operatorname{ad}_{X}$ is skew-adjoint for all $X\in\mathfrak{g}$.

Now, we will see that we can obtain a special class of compact flat manifolds as quotients of Lie groups with a flat invariant metric. We recall first a theorem about the differentiable structure of a quotient manifold $G/\varGamma$ in the case that $\varGamma$ is a closed subgroup of the Lie group $G$.

Theorem 2.11**.**

Let $\varGamma$ be a closed subgroup of a Lie group $G$. The left coset space $G/\varGamma$ is a topological manifold of dimension equal to $\dim G-\dim\varGamma$, and has a unique smooth structure such that the quotient map $\pi:G\to G/\varGamma$ is a smooth submersion.

When $\varGamma$ is a discrete subgroup of a Lie group $G$, hence closed and [math]-dimensional, the quotient map $\pi:G\to G/\varGamma$ is a local diffeomorphism. Let us assume now that $G$ carries a left invariant Riemannian metric. Instead of working with the left coset space, we will consider the right coset space $\varGamma\backslash G$ (note that the theorem above is also valid for the right coset space), to be able to induce a Riemannian metric on $\varGamma\backslash G$ so that $\pi$ results an isometry: Given $g\in G$ and $u,v\in T_{\pi(g)}(\varGamma\backslash G)$, we define

[TABLE]

The metric is well defined since $L_{\gamma}$ is an isometry of $G$ and $\pi\circ L_{\gamma}=\pi$ for all $\gamma\in\varGamma$, where $L_{\gamma}(g)=\gamma g$ for all $g\in G$. It is clear that $\pi$ is a local isometry. In particular, if the metric on $G$ is flat, the induced metric on $\varGamma\backslash G$.

The most interesting case is when $\varGamma$ is a co-compact discrete subgroup of $G$, which is called a lattice of $G$.

Definition 2.12**.**

A solvmanifold $M$ is a compact quotient $\varGamma\backslash G$ where $G$ is a simply connected solvable Lie group and $\varGamma$ is a lattice of $G$. When $G$ is nilpotent, $M$ is said to be a nilmanifold.

Remark 2.13*.*

It is worth mentioning that in the literature, different definitions of solvmanifolds are also considered. For example, L. Auslander defines a solvmanifold as a “homogeneous space of connected solvable Lie group”, [Aus61, p. 398], see also [Aus73a, Aus73b]. We will only consider solvmanifolds as in our definition above, and as a consequence they are always compact, orientable and paralellizable.

Every simply connected solvable Lie group is diffeomorphic to $\mathbb{R}^{n}$ (see for instance [Va]) for some $n\in\mathbb{N}$. This implies that a solvmanifold is aspherical, that is, the higher homotopy groups vanish; and $\pi_{1}(\varGamma\backslash G)\cong\varGamma$. The fundamental group plays an important role in the study of solvmanifolds. Indeed, Mostow’s Theorem show that the solvmanifolds are classified, up to diffeomorphism, by their fundamental groups. A proof of this can be found in [Rag, Theorem 3.6].

Theorem 2.14** (Mostow).**

Let $G_{1}$ and $G_{2}$ be simply connected solvable Lie groups with $\varGamma_{i}$ a lattice in $G_{i}$ for $i=1,2$. If $\phi:\varGamma_{1}\to\varGamma_{2}$ is an isomorphism then there exists a diffeomorphism $\tilde{\phi}:G_{1}\to G_{2}$ such that $\tilde{\phi}|_{\varGamma_{1}}=\phi$ and $\tilde{\phi}(\gamma g)=\phi(\gamma)\tilde{\phi}(g)$ for all $\gamma\in\varGamma_{1},g\in G_{1}$.

Corollary 2.15**.**

Two solvmanifolds with isomorphic fundamental groups are diffeomorphic.

Concerning the construction of solvmanifolds and nilmanifolds, Malcev’s theorem [Ma] asserts that a simply connected nilpotent Lie group $G$ admits a lattice if and only if its Lie algebra has a basis such that the structure constants are rational numbers, so it is not so difficult to construct nilmanifolds. Unfortunately, it is not easy to construct solvmanifolds, since there is not a general method to find lattices in a given solvable Lie group. It is known ([Mi]) that a necessary condition for a solvable Lie group to admit lattices is to be unimodular, but in general there are no known sufficient conditions.

A special class of solvmanifolds that we are going to consider later are those obtained as quotients of almost abelian Lie groups, which we will call almost abelian solvmanifolds.

Definition 2.16**.**

A non-abelian Lie algebra $\mathfrak{g}$ is said to be almost abelian if $\mathfrak{g}$ has a codimension 1 abelian ideal, that is $\mathfrak{g}=\mathbb{R}x\ltimes_{\operatorname{ad}_{x}}\mathbb{R}^{n}$ for some $n\in\mathbb{N}$.

It is clear that an almost abelian Lie algebra is solvable and it is nilpotent if and only if $\operatorname{ad}_{x}|_{\mathbb{R}^{n}}$ is nilpotent. Moreover, if we denote by $\mathfrak{g}_{A}$ the almost abelian Lie algebra with $\operatorname{ad}_{x}|_{\mathbb{R}^{n}}=A$, it can be proved that $\mathfrak{g}_{A}\cong\mathfrak{g}_{B}$ if and only if there exist $0\neq\lambda\in\mathbb{R}$ and $C\in{\sf GL}(n,\mathbb{R})$ such that $B=\lambda CAC^{-1}$.

The simply connected Lie group $G$ corresponding to an almost abelian Lie algebra $\mathfrak{g}=\mathbb{R}x\ltimes_{\operatorname{ad}_{x}}\mathbb{R}^{n}$, is called an almost abelian Lie group, and can be written as $G=\mathbb{R}\ltimes_{\varphi}\mathbb{R}^{n}$, where $\varphi(t)=\exp(t\operatorname{ad}_{x})$.

A great advantage of almost abelian Lie groups is that there exists a criterion to determine whether they admit lattices or not (see [B]).

Proposition 2.17**.**

Let $G=\mathbb{R}\ltimes_{\varphi}\mathbb{R}^{m}$ be an almost abelian Lie group. Then $G$ admits a lattice $\varGamma$ if and only if there exists $t_{0}\neq 0$ such that $\varphi(t_{0})$ is conjugate to an integer matrix invertible in $\mathbb{Z}$. In this case, the lattice $\varGamma$ is given by

[TABLE]

where $P^{-1}{\varphi(t_{0})}P$ is an integer matrix.

3. Holonomy of flat solvmanifolds

In this section we are going to consider solvmanifolds $\varGamma\backslash G$ equipped with a flat metric induced by a flat left invariant metric on $G$, using the characterization given by Milnor mentioned above.

We do not assume that $G$ acts by isometries on $\varGamma\backslash G$. In fact, it is well known that if $M$ is a flat Riemannian manifold with a transitive group of isometries, then $M\cong T^{m}\times\mathbb{R}^{k}$ for some $k,m\in\mathbb{N}$ (see [Wo]).

In general, if a simply connected Lie group $G$ admits a flat left invariant metric, then $G$ is isometric as a Riemannian manifold to $\mathbb{R}^{n}$ with the usual metric. We can identify $\varGamma$ with a subgroup of isometries of $\mathbb{R}^{n}$ as follows:

Let $\psi:G\to\mathbb{R}^{n}$ an isometry. Define a map $i:G\to\operatorname{Iso}(\mathbb{R}^{n})$ by $i(g)=\tau_{g}$, where $\tau_{g}=\psi\circ L_{g}\circ\psi^{-1}$. It is clear that $i$ is an injective group homomorphism, and therefore we can identify $\varGamma$ with its image and consider $\varGamma\subset\operatorname{Iso}(\mathbb{R}^{n})$.

The solvmanifold $\varGamma\backslash G$ is isometric to $\mathbb{R}^{n}/\varGamma$ by considering the map $\phi([g])=[\psi(g)]$. Observe that $\phi$ is well defined since

[TABLE]

Furthermore it is injective, since if $[\psi(g)]=[\psi(h)]$ then $\psi(g)=\tau_{\gamma}(\psi(h))$, which implies $\psi(g)=\psi(\gamma h)$. Thus, $g=\gamma h$, or equivalently, $[g]=[h]$. The surjectivity is clear, and as $\phi$ is a composition of local isometries and bijective, $\phi$ results an isometry. Therefore $\varGamma$ is a Bieberbach subgroup of $\operatorname{Iso}(\mathbb{R}^{n})$ and $\operatorname{Hol}(\mathbb{R}^{n}/\varGamma)\cong r(\varGamma)$. We recall that $r(\varGamma)$ is identified with the finite group $\varGamma/\Lambda$ where $\Lambda$ is the maximal normal abelian subgroup of $\varGamma$. In conclusion,

[TABLE]