1 Introduction

Riemannian locally symmetric spaces were characterized by Cartan as those with parallel curvature with respect to the Levi–Civita connection. Homogeneous manifolds constitute a rich class of spaces to study Riemannian geometry. Ambrose and Singer [1] extended the above characterization to this kind of Riemannian manifolds showing that a connected, complete, and simply connected Riemannian manifold (M, g) is homogeneous if and only if there exists a (1, 2)-tensor field T on M such that

$$\begin{aligned} { \widetilde{\nabla }g=0,} \qquad { \widetilde{\nabla }R=0,} \qquad {\widetilde{\nabla }T=0,} \end{aligned}$$
(1)

where \(\widetilde{\nabla }\) is the Ambrose–Singer connection given by \(\widetilde{\nabla }=\nabla -T\), \(\nabla \) is the Levi–Civita connection of the metric g, and R denotes the Riemannian curvature tensor for which we adopt the sign convention \(R(X,Y) = \nabla _{[X,Y]} - \left[ \nabla _X, \nabla _Y\right] \).

The difference tensor field T is said to be a homogeneous structure on M. T will also denote the associated tensor field of type (0, 3) given by \(T(X,Y,Z)=g(T(X,Y),Z)\). Conditions (1) were further investigated by Tricerri and Vanhecke [11], who considered the space \(\mathcal {T}(\mathcal {V})\) of such tensor fields on a vector space \((\mathcal {V},\langle ,\rangle )\) and decomposed it into three irreducible components under the action of the orthogonal group as \(\mathcal {T}(\mathcal {V}) =\mathcal {T}_1(\mathcal {V}) \oplus \mathcal {T}_2(\mathcal {V}) \oplus \mathcal {T}_3(\mathcal {V})\). The subspaces of such decomposition are given as follows

$$\begin{aligned} \mathcal {T}_1(\mathcal {V})= & {} \{ T\in \mathcal {T}(\mathcal {V}) :T(x,y,z)=\langle x,y\rangle \varphi (z)-\langle x,z\rangle \varphi (y)\},\\ \mathcal {T}_2(\mathcal {V})= & {} \{ T\in \mathcal {T}(\mathcal {V}) :c_{12}(T)=0, \,\, \sigma _{x,y,z}T(x,y,z)=0\},\\ \mathcal {T}_3(\mathcal {V})= & {} \{ T\in \mathcal {T}(\mathcal {V}) :T(x,y,z)+T(y,x,z)=0\}, \end{aligned}$$

where \(c_{12}({T})\) denotes the contraction, \(c_{12}(T)(z)=\sum _i T(e_i,e_i,z)\) for an arbitrary orthonormal basis \(\{ e_i\}\) of \(\mathcal {V}\), \(\sigma _{x,y,z}\) denotes the cyclic sum with respect to x, y, z, and \(\varphi \in \mathcal {V}^*\). The projections of a homogeneous structure T on each of these subspaces are given by

$$\begin{aligned} p_1(T)(x,y,z)= & {} \frac{1}{2}\langle x,y\rangle c_{12}(T)(z)-\frac{1}{2}\langle x,z\rangle c_{12}(T)(y),\nonumber \\ p_3(T)(x,y,z)= & {} \frac{1}{3}\sigma _{x,y,z}T(x,y,z),\nonumber \\ p_2(T)(x,y,z)= & {} \left( T-p_1(T)-p_3(T)\right) (x,y,z). \end{aligned}$$
(2)

Homogeneous manifolds admitting a homogeneous structure in one of the eight different classes induced by the above decomposition have been extensively studied in the literature. It was shown in [11] that naturally reductive spaces correspond to non-vanishing homogeneous structures of type \(\mathcal {T}_3\) and that a Riemannian manifold admits a non-vanishing structure of type \(\mathcal {T}_1\) if and only if it is locally isometric to the real hyperbolic space. The later also holds true for homogeneous structures of type \(\mathcal {T}_1\oplus \mathcal {T}_3\), \(T\notin \mathcal {T}_1\) and \(T\notin \mathcal {T}_3\), in dimension greater than three, as shown in [8]. Riemannian manifolds of dimension less or equal to four admitting a homogeneous structure of type \(\mathcal {T}_2\) were described in [5] (see also [2]). Homogeneous structures in the class \(\mathcal {T}_1\oplus \mathcal {T}_2\) in dimension less or equal to four were described in [3], and those in this class whose fundamental 1-form is closed were investigated in [9]. It was shown in [5] that a three-dimensional non-symmetric space admitting a homogeneous structure of type \(\mathcal {T}_3\) also admits a \(\mathcal {T}_2\)-structure.

In dimension two \(\mathcal {T}(\mathcal {V})=\mathcal {T}_1(\mathcal {V})\), and hence a surface admits a non-zero homogeneous structure if and only if it is isometric to the hyperbolic plane. Dimension three is particularly relevant in the study of homogeneous spaces. First of all, it is the lowest possible dimension admitting locally homogeneous metrics which are not locally symmetric and, secondly, any three-dimensional homogeneous manifold is either symmetric or locally isometric to a Lie group endowed with a left-invariant metric [10].

The special case when (M, g) is a Lie group G equipped with a left-invariant metric \(\langle ,\rangle \) is of special interest for our purposes. Let \(T^\nabla \) be the canonical homogeneous structure defined by

$$\begin{aligned} 2\langle T^\nabla (X,Y),Z\rangle =\langle [X,Y],Z\rangle -\langle [Y,Z],X\rangle +\langle [Z,X],Y\rangle , \end{aligned}$$

for left-invariant vector fields X, Y and Z. Then the corresponding Ambrose-Singer connection \(\widetilde{\nabla }=\nabla -T^\nabla \) satisfies \(\widetilde{\nabla }_XY=0\) for left-invariant vector fields. This structure is equivalent to the description \(G=G/\{ e\}\), which corresponds to the action \(G\times G\rightarrow G\).

On the basis of the above, the aim of this work is to clarify the classification of the homogeneous Riemannian structures in dimension three, giving all the possible ones in the non-symmetric case. The following result characterizes the non-symmetric Lie groups admitting more than one homogeneous structure.

Theorem 1.1

A non-symmetric simply connected three-dimensional Riemannian Lie group admits a homogeneous structure different from the canonical one if and only if it admits a naturally reductive homogeneous structure. Moreover, in such a case, it admits exactly a one-parameter family of homogeneous structures.

The explicit description of all homogeneous structures on non-symmetric Lie groups is given in Theorems 1.2 and 1.3 by considering separately the unimodular and non-unimodular cases.

We recall that a three-dimensional complete and simply connected manifold is naturally reductive if and only if it admits a non-vanishing homogeneous structure of type \(\mathcal {T}_3\). In this case (M, g) is a real space form \(\mathbb {R}^3\), \(\mathbb {S}^3\) or \(\mathbb {H}^3\), or it is isometric either to the special unitary group SU(2), or to the universal cover of \(SL(2, \mathbb {R})\) or to the 3-dimensional Heisenberg group \(H^3\), endowed with a suitable left-invariant metric described in terms of the Lie algebra (up to rotations) by

$$\begin{aligned} H^3: [e_1,e_2]= & {} \lambda e_3,\,\,\lambda \ne 0,\nonumber \\ SU(2): [e_1,e_2]= & {} \mu e_3,\,\,[e_2,e_3] = \lambda e_1,\,\,[e_3,e_1] = \lambda e_2, \lambda \,\mu >0,\nonumber \\ SL(2,\mathbb {R}): [e_1,e_2]= & {} \mu e_3,\,\,[e_2,e_3] = \lambda e_1,\,\,[e_3,e_1] = \lambda e_2, \lambda \,\mu <0, \end{aligned}$$
(3)

where \(\{e_1,e_2,e_3\}\) is an orthonormal basis (see [11]).

In this way, (M, g) is naturally reductive if and only if it is isometric to a Lie group endowed with a left-invariant metric whose isometry group is at least four-dimensional.

Theorem 1.1 is thus connected to the following theorem by Meeks and Perez (see [6]): a simply connected, 3-dimensional Lie group with a left-invariant metric \((G_1,\langle ,\rangle _1)\) is isometric to a second Lie group \((G_2,\langle ,\rangle _2)\) such that is not isomorphic to \(G_1\) if and only if its isometry group has dimension at least 4.

1.1 Summary of results

We study the unimodular and non-unimodular cases separately. The unimodular case is dealt with in Sect. 2, and the non-unimodular case is considered in Sect. 3. We show that the homogeneous Riemannian structures on a non-symmetric three-dimensional Lie group G equipped with a left-invariant metric are given as follows, from where the proof of Theorem 1.1 is obtained at once.

1.1.1 Unimodular Lie groups

Left-invariant Riemannian metrics \(\langle ,\rangle \) on unimodular Lie groups G were described by Milnor (see [7]) in terms of parameters \((\lambda _1,\lambda _2,\lambda _3)\), so that the Lie algebra becomes

$$\begin{aligned} {[}e_1,e_2] = \lambda _3 e_3,\qquad [e_1,e_3] = -\lambda _2 e_2,\qquad [e_2,e_3] = \lambda _1 e_1, \end{aligned}$$

where \(\{e_1,e_2,e_3\}\) is an orthonormal basis. It now follows that \((G,\langle ,\rangle )\) is symmetric if and only if \(\lambda _1=0\) and \(\lambda _2=\lambda _3\) (up to rotations), in which case it is flat, or \(\lambda _1=\lambda _2=\lambda _3\ne 0\), and the sectional curvature is constant and positive. Now one has

Theorem 1.2

Let \((G,\langle ,\rangle )\) be a unimodular Lie group equipped with a left-invariant non-symmetric Riemannian metric. Then there are two mutually excluding cases.

  1. (i)

    The three structure constants \(\lambda _1,\lambda _2,\lambda _3\) are different and the only homogeneous structure is the canonical one, given by

    $$\begin{aligned} T^\nabla= & {} - (\lambda _1-\lambda _2-\lambda _3)e^1\otimes (e^2\wedge e^3) - (\lambda _1-\lambda _2+\lambda _3)e^2\otimes (e^1\wedge e^3)\\{} & {} + (\lambda _1+\lambda _2-\lambda _3) e^3\otimes (e^1\wedge e^2). \end{aligned}$$

    The canonical homogeneous structure is of type \(\mathcal {T}_2\) if \(\lambda _1+\lambda _2+\lambda _3=0\) (see also [3]) and it is of type \(\mathcal {T}_2\oplus \mathcal {T}_3\) otherwise.

  2. (ii)

    Up to a rotation, the structure constants \(\lambda _1=\lambda _2\ne \lambda _3\), \(\lambda _3\ne 0\) and there exists a one-parameter family of homogeneous structures

    $$\begin{aligned} T= \lambda _3e^1\otimes (e^2\wedge e^3) - \lambda _3e^2\otimes (e^1\wedge e^3) + 2\kappa e^3\otimes (e^1\wedge e^2), \quad \kappa \in \mathbb {R}, \end{aligned}$$

    which corresponds to the canonical structure for \(\kappa =\frac{1}{2}(2\lambda _1-\lambda _3)\). Moreover, it is of type \(\mathcal {T}_2\) if \(\kappa =-\lambda _3\), of type \(\mathcal {T}_3\) if \(\kappa =\frac{1}{2}\lambda _3\), and of type \(\mathcal {T}_2\oplus \mathcal {T}_3\) otherwise.

Unimodular Lie groups in Theorem 1.2-(ii) correspond to SU(2), \(\widetilde{SL}(2,\mathbb {R})\) and \(H^3\) with left-invariant metric as in (3), which contains the case of the homogeneous structures on Berger spheres previously considered in [4].

1.1.2 Non-unimodular Lie groups

Non-unimodular Riemannian Lie groups \((G,\langle ,\rangle )\) are semi-direct extensions \(\mathbb {R}\ltimes \mathbb {R}^2\) of the Abelian group. It was shown in [7] that there exist an orthonormal basis \(\{e_1,e_2,e_3\}\) so that

$$\begin{aligned} {[}e_1,e_2] = \alpha e_2+\beta e_3,\qquad [e_1,e_3] =\gamma e_2+\delta e_3,\qquad [e_2,e_3] = 0, \end{aligned}$$

where the trace of the endomorphism determining the semi-direct extension \(\alpha +\delta \ne 0\). Moreover, one may rotate the orthonormal basis \(\{e_2,e_3\}\) of the unimodular kernel to assume that their images by the endomorphism are orthogonal, i.e., \(\alpha \gamma +\beta \delta =0\). The Riemannian Lie group \((G,\langle ,\rangle )\) is symmetric if and only if \(\beta =\delta =\gamma =0\) (up to the isometry \(e_2\mapsto e_3\)), and so it is isometric to \(\mathbb {R}\times \mathbb {H}^2(-\alpha ^2)\), or if \(\alpha =\delta \ne 0\) and \(\gamma =-\beta \) (in which case it is a space of constant sectional curvature \(\mathbb {H}^3(-\delta ^2)\)). Now one has the following.

Theorem 1.3

Let \((G,\langle ,\rangle )\) be a non-unimodular Lie group equipped with a left-invariant non-symmetric Riemannian metric. Then there are two mutually excluding cases.

  1. (i)

    If \(\delta =\gamma =0\), \(\alpha \beta \ne 0\), then the homogeneous structures are given by:

    1. (i.a)

      The one-parameter family

      $$\begin{aligned} T=\beta e^1\otimes (e^2\wedge e^3) - \beta e^2\otimes (e^1\wedge e^3) + 2\kappa e^3\otimes (e^1\wedge e^2), \quad \kappa \in \mathbb {R}. \end{aligned}$$

      In this case the homogeneous structure is of type \(\mathcal {T}_2\) if \(\kappa =-\beta \), of type \(\mathcal {T}_3\) if \(\kappa =\frac{1}{2}\beta \) or of type \(\mathcal {T}_2\oplus \mathcal {T}_3\) otherwise.

    2. (i.b)

      The canonical homogeneous structure

      $$\begin{aligned} T^\nabla =\beta e^1\otimes (e^2\wedge e^3) -2\alpha e^2\otimes (e^1\wedge e^2) -\beta e^2\otimes (e^1\wedge e^3) - \beta e^3\otimes (e^1\wedge e^2), \end{aligned}$$

      which is of the generic type \(\mathcal {T}_1\oplus \mathcal {T}_2\oplus \mathcal {T}_3\).

  2. (ii)

    If \(\delta \alpha \ne 0\), \(\beta =-\frac{\alpha \gamma }{\delta }\) and \(\alpha \ne \delta \), then the only homogeneous structure is the canonical one, given by

    $$\begin{aligned} T^\nabla= & {} -\tfrac{(\alpha +\delta )\gamma }{\delta } e^1\otimes (e^2\wedge e^3) -2\alpha e^2\otimes (e^1\wedge e^2) +\tfrac{(\alpha -\delta )\gamma }{\delta } e^2\otimes (e^1\wedge e^3)\\{} & {} + \tfrac{(\alpha -\delta )\gamma }{\delta } e^3\otimes (e^1\wedge e^2) -2\delta e^3\otimes (e^1\wedge e^3), \end{aligned}$$

    which is of type \(\mathcal {T}_1\oplus \mathcal {T}_2\) if \(\gamma =0\), and of the generic \(\mathcal {T}_1\oplus \mathcal {T}_2\oplus \mathcal {T}_3\) otherwise.

Non-unimodular Lie groups in Theorem 1.3 are semi-direct extensions \(\mathbb {R}\ltimes \mathbb {R}^2\) of the Abelian Lie group determined by an endomorphism \(-{\text {ad}}(e_1)\). Assertion (i) in Theorem 1.3 corresponds to the special situation \({\text {det}}{\text {ad}}(e_1)=0\), and they are isometric (although not isomorphically isometric) to a left-invariant metric on \(\widetilde{SL}(2,\mathbb {R})\) as in (3) corresponding to Theorem 1.2-(ii) (cf. [6, 11]). We emphasize that isometries between Riemannian Lie groups need not preserve the Lie group structure, since they are not necessarily realized by group isomorphisms, as evidenced in the above-mentioned situation. On the contrary, Lie groups in Theorem 1.3-(ii) correspond to the generic situation, where one may always specialize the orthonormal basis \(\{e_2,e_3\}\) to be given by eigenvectors of the self-adjoint part of \({\text {ad}}(e_1)\) (cf. [7]).

Non-symmetric simply connected homogeneous three-dimensional Riemannian manifolds with four-dimensional isometry group are isometric to the unitary group SU(2), the universal cover of \(SL(2,\mathbb {R})\), or the Heisenberg group with the special metrics (3). It follows from the description of homogeneous structures in Theorems 1.2 and 1.3 that (see also [11]):

A non-symmetric three-dimensional Riemannian Lie group admits a homogeneous structure different from the canonical one if and only if the isometry group is four-dimensional.

Remark 1.4

A more conceptual proof of this last statement can be summarized as follows. For any three-dimensional Lie groups \((G_1,\langle ,\rangle _1)\) and \((G_2,\langle ,\rangle _2)\) equipped with a left-invariant Riemannian metric, it follows from Theorems 1.2 and 1.3 that the infinitesimal models associated to their canonical homogeneous structures are isomorphic if and only if \((G_1,\langle ,\rangle _1)\) and \((G_2,\langle ,\rangle _2)\) are isomorphically isometric (see [2]). Besides, any non-symmetric homogeneous three-manifold with four-dimensional isometry group admits more than one homogeneous structure. It follows from the work in [6, 10] that a homogeneous three-manifold with three-dimensional isometry group is isometric to a unique Riemannian Lie group, in which case any homogeneous structure is isomorphic to the canonical one.

2 Homogeneous structures on non-symmetric unimodular Lie groups

Following [7], if \(\mathfrak {g}\) is unimodular then there exists an orthonormal basis \(\{e_1,e_2,e_3\}\) of \(\mathfrak {g}\) such that

$$\begin{aligned} {[}e_1,e_2] = \lambda _3 e_3,\qquad [e_1,e_3] = -\lambda _2 e_2,\qquad [e_2,e_3] = \lambda _1 e_1. \end{aligned}$$

A unimodular Lie group corresponding to a Lie algebra as above is locally symmetric if and only if the eigenvalues of the structure operator satisfy \(\lambda _1=\lambda _2=\lambda _3\ne 0\) (in which case the sectional curvature is constant and positive) or, up to a rotation, one has \(\lambda _1=0\) and \(\lambda _2=\lambda _3\) (in which case the metric is flat).

Let T be a (0, 3)-tensor field so that the connection \(\widetilde{\nabla }=\nabla -T\) makes the metric tensor parallel, i.e., \(T_{xyz} + T_{xzy}=0\) for x, y, \(z\in \mathfrak {g}\). Denoting by \(\{e^1,e^2,e^3\}\) the dual basis of \(\{e_1,e_2,e_3\}\), then the tensor field T can be written as \(T=2\sum _i\sum _{j<k} T_{ijk} e^i\otimes (e^j\wedge e^k)\). Therefore, the non-zero components of the connection \(\widetilde{\nabla }=\nabla -T\) are given by

$$\begin{aligned} \widetilde{\nabla }_{112}= & {} -T_{112}, \widetilde{\nabla }_{223}=-T_{223}, \widetilde{\nabla }_{123}=-\frac{1}{2}(\lambda _1-\lambda _2-\lambda _3)-T_{123},\nonumber \\ \widetilde{\nabla }_{113}= & {} -T_{113}, \widetilde{\nabla }_{313}=-T_{313}, \widetilde{\nabla }_{213}=-\frac{1}{2}(\lambda _1-\lambda _2+\lambda _3)-T_{213},\nonumber \\ \widetilde{\nabla }_{212}= & {} -T_{212}, \widetilde{\nabla }_{323}=-T_{323}, \widetilde{\nabla }_{312}= \frac{1}{2}(\lambda _1+\lambda _2-\lambda _3)-T_{312}, \end{aligned}$$
(4)

while the (0, 4)-curvature tensor field is determined by

$$\begin{aligned} R_{1212}= & {} \frac{1}{4}\left( (\lambda _1-\lambda _2)^2-3\lambda _3^2+2(\lambda _1+\lambda _2)\lambda _3\right) ,\nonumber \\ R_{1313}= & {} \frac{1}{4}\left( (\lambda _1-\lambda _3)^2-3\lambda _2^2+2(\lambda _1+\lambda _3)\lambda _2\right) ,\nonumber \\ R_{2323}= & {} \frac{1}{4}\left( (\lambda _2-\lambda _3)^2-3\lambda _1^2+2(\lambda _2+\lambda _3)\lambda _1\right) . \end{aligned}$$
(5)

Let \(\mathfrak {R}_{ikj\ell ;r}=(\widetilde{\nabla }_{e_r} R)(e_i,e_j,e_k,e_\ell )\). A straightforward calculation using Eqs. (4) and (5) shows that the condition \(\widetilde{\nabla }R=0\) in Eq. (1) is given by

$$\begin{aligned} 2\,\mathfrak {R}_{1213;1}= & {} (\lambda _1-\lambda _2-\lambda _3)(\lambda _2-\lambda _3)(\lambda _1-\lambda _2-\lambda _3+2T_{123})=0,\nonumber \\ \mathfrak {R}_{1213;2}= & {} (\lambda _1-\lambda _2-\lambda _3)(\lambda _2-\lambda _3)T_{223}=0,\nonumber \\ \mathfrak {R}_{1213;3}= & {} (\lambda _1-\lambda _2-\lambda _3)(\lambda _2-\lambda _3)T_{323}=0,\nonumber \\ \mathfrak {R}_{1223;1}= & {} (\lambda _1-\lambda _2+\lambda _3)(\lambda _1-\lambda _3)T_{113}=0,\nonumber \\ 2\, \mathfrak {R}_{1223;2}= & {} (\lambda _1-\lambda _2+\lambda _3)(\lambda _1-\lambda _3)(\lambda _1-\lambda _2+\lambda _3+2T_{213})=0,\nonumber \\ \mathfrak {R}_{1223;3}= & {} (\lambda _1-\lambda _2+\lambda _3)(\lambda _1-\lambda _3)T_{313}=0, \nonumber \\ -\mathfrak {R}_{1323;1}= & {} (\lambda _1+\lambda _2-\lambda _3)(\lambda _1-\lambda _2)T_{112}=0,\nonumber \\ -\mathfrak {R}_{1323;2}= & {} (\lambda _1+\lambda _2-\lambda _3)(\lambda _1-\lambda _2)T_{212}=0,\nonumber \\ 2\, \mathfrak {R}_{1323;3}= & {} (\lambda _1+\lambda _2-\lambda _3)(\lambda _1-\lambda _2)(\lambda _1+\lambda _2-\lambda _3-2T_{312})=0. \end{aligned}$$
(6)

Next, depending on the eigenvalues \(\lambda _i\), we are led to the following two possibilities.

2.1 Case of three different eigenvalues

In this case, if \(\lambda _1-\lambda _2-\lambda _3\), \(\lambda _1-\lambda _2+\lambda _3\) and \(\lambda _1+\lambda _2-\lambda _3\) do not vanish, then Eqs. (4) and (6) clearly imply that \(\widetilde{\nabla }_XY=0\) for left-invariant vector fields. Hence, the only homogeneous structure is the canonical one given by \(T_xy=\nabla _xy\), for x, \(y\in \mathfrak {g}\), i.e.,

$$\begin{aligned} T= & {} - (\lambda _1-\lambda _2-\lambda _3)e^1\otimes (e^2\wedge e^3) - (\lambda _1-\lambda _2+\lambda _3)e^2\otimes (e^1\wedge e^3)\nonumber \\{} & {} + (\lambda _1+\lambda _2-\lambda _3) e^3\otimes (e^1\wedge e^2). \end{aligned}$$
(7)

Next we show that the same holds if any of \(\lambda _1-\lambda _2-\lambda _3\), \(\lambda _1-\lambda _2+\lambda _3\) and \(\lambda _1+\lambda _2-\lambda _3\) vanishes. Suppose that \(\lambda _3=\lambda _1-\lambda _2\) (the other two cases are obtained in a completely analogous way). Then, Eq. (6) implies

$$\begin{aligned} T_{112}=T_{113}=T_{212}=T_{313}=0,\quad T_{213}=-\lambda _1+\lambda _2,\quad T_{312}=\lambda _2. \end{aligned}$$
(8)

Let \(\mathfrak {T}_{ijk;r}=(\widetilde{\nabla }_{e_r} T)(e_i,e_j,e_k)\). A straightforward calculation using Eqs. (4) and (8) shows that the condition \(\widetilde{\nabla }T=0\) in Eq. (1) reduces to

$$\begin{aligned} \mathfrak {T}_{313;r}= & {} -\mathfrak {T}_{212;r}=(\lambda _1-2\lambda _2)T_{r23}=0, \\ \mathfrak {T}_{223;r}= & {} T_{r23} T_{323}=0,\\ \mathfrak {T}_{323;r}= & {} -T_{r23} T_{223}=0, \end{aligned}$$

or, equivalently, \(T_{123}=T_{223}=T_{323}=0\). Thus, the only homogeneous structure is given by \(T=-2(\lambda _1-\lambda _2)e^2\otimes (e^1\wedge e^3)+2\lambda _2 e^3\otimes (e^1\wedge e^2)\), which corresponds to the structure given by Eq. (7) for \(\lambda _3=\lambda _1-\lambda _2\).

Finally, the projections of the homogeneous structure given in Eq. (7) are obtained by a direct calculation. In particular, \(p_1(T)=0\) and

$$\begin{aligned} p_2(T)= & {} -\frac{2}{3} (2\lambda _1-\lambda _2-\lambda _3)e^1\otimes (e^2\wedge e^3) -\frac{2}{3} (\lambda _1-2\lambda _2+\lambda _3)e^2\otimes (e^1\wedge e^3)\\{} & {} + \frac{2}{3} (\lambda _1+\lambda _2-2\lambda _3) e^3\otimes (e^1\wedge e^2),\\ p_3(T)= & {} \frac{1}{3} (\lambda _1+\lambda _2+\lambda _3) \left( e^1\otimes (e^2\wedge e^3)- e^2\otimes (e^1\wedge e^3) + e^3\otimes (e^1\wedge e^2) \right) . \end{aligned}$$

2.2 Case of two different eigenvalues

In this case, without loss of generality, we can assume \(\lambda _1=\lambda _2\ne \lambda _3\). Moreover, \(\lambda _3\ne 0\) since the space would be locally symmetric otherwise. Thus, Eq. (6) implies

$$\begin{aligned} T_{113}=T_{223}=T_{313}=T_{323}=0,\quad T_{123}=-T_{213}=\frac{\lambda _3}{2}. \end{aligned}$$
(9)

Let \(\mathfrak {T}_{ijk;r}=(\widetilde{\nabla }_{e_r} T)(e_i,e_j,e_k)\). A straightforward calculation using Eqs. (4) and (9) shows that the condition \(\widetilde{\nabla }T=0\) in Eq. (1) reduces to

$$\begin{aligned} \mathfrak {T}_{112;r}= & {} T_{r12} T_{212}=0,\,\, (r=1,2),\,\, \mathfrak {T}_{112;3}=-\frac{1}{2}(2\lambda _1-\lambda _3-2T_{312}) T_{212}=0,\\ \mathfrak {T}_{212;r}= & {} -T_{r12} T_{112}=0,\,\, (r=1,2), \,\, \quad \mathfrak {T}_{212;3}=\frac{1}{2}(2\lambda _1-\lambda _3-2T_{312}) T_{112}=0, \end{aligned}$$

or, equivalently, \(T_{112}=T_{212} =0\). Thus, we obtain a one-parameter family of homogeneous structures given by

$$\begin{aligned} T= \lambda _3e^1\otimes (e^2\wedge e^3) - \lambda _3e^2\otimes (e^1\wedge e^3) + 2\kappa e^3\otimes (e^1\wedge e^2), \quad \kappa \in \mathbb {R}. \end{aligned}$$

In the particular case where \(\kappa =\frac{1}{2}(2\lambda _1-\lambda _3)\) it corresponds to the canonical structure. Finally, a direct calculation shows that the projections of these structures are such that \(p_1(T)=0\) and

$$\begin{aligned} p_2(T)= & {} \frac{1}{3} (\lambda _3-2\kappa ) \left( e^1\otimes (e^2\wedge e^3) - e^2\otimes (e^1\wedge e^3) - 2 e^3\otimes (e^1\wedge e^2) \right) ,\\ p_3(T)= & {} \frac{2}{3} (\lambda _3+\kappa ) \left( e^1\otimes (e^2\wedge e^3)- e^2\otimes (e^1\wedge e^3) + e^3\otimes (e^1\wedge e^2) \right) . \end{aligned}$$

3 Homogeneous structures on non-symmetric non-unimodular Lie groups

If \(\mathfrak {g}\) is non-unimodular then there exists an orthonormal basis \(\{e_1,e_2,e_3\}\) of \(\mathfrak {g}\) such that (see [7])

$$\begin{aligned} {[}e_1,e_2] = \alpha e_2+\beta e_3,\qquad [e_1,e_3] =\gamma e_2+\delta e_3,\qquad [e_2,e_3] = 0, \end{aligned}$$

where \(\alpha +\delta \ne 0\) and \(\alpha \gamma +\beta \delta =0\).

A straightforward calculation shows that a non-unimodular Lie group corresponding to a Lie algebra above is locally symmetric if and only if it is of constant negative sectional curvature (which corresponds to the cases when \({\text {ad}}(e_1)\) is a multiple of the identity or it has complex eigenvalues), or it is locally isometric to a product \(\mathbb {R}\times N(c)\), where N(c) is a surface of constant negative sectional curvature (if \({\text {ad}}(e_1)\) is of rank-one and \(\{e_2,e_3\}\) is an orthonormal basis of eigenvectors).

Using the same notation as in the previous section, the non-zero components of the connection \(\widetilde{\nabla }=\nabla -T\) are given by

$$\begin{aligned} \widetilde{\nabla }_{112}= & {} -T_{112}, \quad \widetilde{\nabla }_{223}=-T_{223}, \quad \widetilde{\nabla }_{123}=\frac{1}{2}(\beta -\gamma -2T_{123}),\nonumber \\ \widetilde{\nabla }_{113}= & {} -T_{113}, \quad \widetilde{\nabla }_{313}=-\delta -T_{313}, \quad \widetilde{\nabla }_{213}=-\frac{1}{2}(\beta +\gamma +2T_{213}),\nonumber \\ \widetilde{\nabla }_{212}= & {} -\alpha -T_{212}, \quad \widetilde{\nabla }_{323}=-T_{323}, \quad \widetilde{\nabla }_{312}=-\frac{1}{2}(\beta +\gamma +2T_{312}), \end{aligned}$$
(10)

while the (0, 4)-curvature tensor field is determined by

$$\begin{aligned} R_{1212}= & {} -\frac{1}{4}\left( 4\alpha ^2+3\beta ^2-\gamma ^2+2\beta \gamma \right) , \nonumber \\ R_{1313}= & {} \frac{1}{4}\left( \beta ^2-3\gamma ^2-4\delta ^2-2\beta \gamma \right) , \nonumber \\ R_{2323}= & {} \frac{1}{4}\left( \beta ^2+\gamma ^2-4\alpha \delta +2\beta \gamma \right) . \end{aligned}$$
(11)

As in the previous section, set \(\mathfrak {R}_{ikj\ell ;r}=(\widetilde{\nabla }_{e_r} R)(e_i,e_j,e_k,e_\ell )\). Equations (10) and (11) imply that the condition \(\widetilde{\nabla }R=0\) in Eq. (1) is given by

$$\begin{aligned} -2\, \mathfrak {R}_{1213;1}= & {} (\alpha ^2+\beta ^2-\gamma ^2-\delta ^2)(\beta -\gamma -2T_{123})=0,\nonumber \\ \mathfrak {R}_{1213;2}= & {} (\alpha ^2+\beta ^2-\gamma ^2-\delta ^2) T_{223}=0,\nonumber \\ \mathfrak {R}_{1213;3}= & {} (\alpha ^2+\beta ^2-\gamma ^2-\delta ^2) T_{323}=0,\nonumber \\ -\mathfrak {R}_{1223;1}= & {} (\alpha ^2+\beta ^2-\alpha \delta +\beta \gamma )T_{113}=0,\nonumber \\ -2\,\mathfrak {R}_{1223;2}= & {} (\alpha ^2+\beta ^2-\alpha \delta +\beta \gamma )(\beta +\gamma +2T_{213}) =0,\nonumber \\ -\mathfrak {R}_{1223;3}= & {} (\alpha ^2+\beta ^2-\alpha \delta +\beta \gamma )(\delta +T_{313}) =0,\nonumber \\ \mathfrak {R}_{1323;1}= & {} (\gamma ^2+\delta ^2-\alpha \delta +\beta \gamma )T_{112} =0,\nonumber \\ \mathfrak {R}_{1323;2}= & {} (\gamma ^2+\delta ^2-\alpha \delta +\beta \gamma )(\alpha +T_{212}) =0, \nonumber \\ 2\,\mathfrak {R}_{1323;3}= & {} (\gamma ^2+\delta ^2-\alpha \delta +\beta \gamma ) (\beta +\gamma +2T_{312})=0. \end{aligned}$$
(12)

Next we analyze the cases \(\delta =0\) and \(\delta \ne 0\) separately.

3.1 Case \(\delta =0\)

Since \(\alpha +\delta \ne 0\) and \(\alpha \gamma +\beta \delta =0\), in this case \(\gamma =0\) and \(\alpha \ne 0\). Moreover, \(\beta \ne 0\) because the space is assumed not to be locally symmetric. Thus, Eq. (12) reduces to

$$\begin{aligned} T_{113}=T_{223}=T_{313}=T_{323}=0,\quad T_{123}=-T_{213}=\frac{\beta }{2} . \end{aligned}$$
(13)

Following the notation in the previous section, we set \(\mathfrak {T}_{ijk;r}=(\widetilde{\nabla }_{e_r} T)(e_i,e_j,e_k)\). A straightforward calculation using Eqs. (10) and (13) shows that the condition \(\widetilde{\nabla }T=0\) in Eq. (1) reduces to

$$\begin{aligned} \mathfrak {T}_{112;1}= & {} T_{112} T_{212}=0, \quad -\mathfrak {T}_{212;1} = (T_{112})^2=0,\\ \mathfrak {T}_{112;2}= & {} (\alpha +T_{212})T_{212}=0, \quad -\mathfrak {T}_{212;2} = (\alpha +T_{212})T_{112}=0,\\ 2\,\mathfrak {T}_{112;3}= & {} (\beta +2T_{312})T_{212}=0,\quad -2\,\mathfrak {T}_{212;3} = (\beta +2T_{312})T_{112}=0. \end{aligned}$$

Hence, \(T_{112}=0\) and either \(T_{212}=0\) or \(T_{212}=-\alpha \), \(T_{312}=-\frac{\beta }{2}\).

If \(T_{112}=T_{212}=0\) we obtain a one-parameter family of homogeneous structures given by

$$\begin{aligned} T=\beta e^1\otimes (e^2\wedge e^3) - \beta e^2\otimes (e^1\wedge e^3) + 2\kappa e^3\otimes (e^1\wedge e^2), \quad \kappa \in \mathbb {R}. \end{aligned}$$

The projections of these homogeneous structures are obtained by a direct calculation. In particular, \(p_1(T)=0\) and

$$\begin{aligned} p_2(T)= & {} \frac{1}{3} (\beta -2\kappa ) \left( e^1\otimes (e^2\wedge e^3) - e^2\otimes (e^1\wedge e^3) - 2 e^3\otimes (e^1\wedge e^2) \right) ,\\ p_3(T)= & {} \frac{2}{3} (\beta +\kappa ) \left( e^1\otimes (e^2\wedge e^3)- e^2\otimes (e^1\wedge e^3) + e^3\otimes (e^1\wedge e^2) \right) . \end{aligned}$$

If \(T_{112}=0\) and \(T_{212}=-\alpha \), \(T_{312}=-\frac{\beta }{2}\), then Eqs. (10) and (13) clearly imply that \(\widetilde{\nabla }_XY=0\) for left-invariant vector fields. Therefore, the only homogeneous structure is the canonical one, given by

$$\begin{aligned} T=\beta e^1\otimes (e^2\wedge e^3) -2\alpha e^2\otimes (e^1\wedge e^2) -\beta e^2\otimes (e^1\wedge e^3) - \beta e^3\otimes (e^1\wedge e^2). \end{aligned}$$

In this case, the projections of this homogeneous structure are given by

$$\begin{aligned} p_1(T)= & {} -\alpha \left( e^2\otimes (e^1\wedge e^2) + e^3\otimes (e^1\wedge e^3) \right) ,\\ p_2(T)= & {} \frac{2}{3}\beta e^1\otimes (e^2\wedge e^3) - \alpha e^2\otimes (e^1\wedge e^2) -\frac{2}{3}\beta e^2\otimes (e^1\wedge e^3)\\{} & {} - \frac{4}{3}\beta e^3\otimes (e^1\wedge e^2) + \alpha e^3\otimes (e^1\wedge e^3),\\ p_3(T)= & {} \frac{1}{3}\beta \left( e^1\otimes (e^2\wedge e^3) - e^2\otimes (e^1\wedge e^3) + e^3\otimes (e^1\wedge e^2) \right) . \end{aligned}$$

3.2 Case \(\delta \ne 0\)

In this case, since \(\alpha \gamma +\beta \delta =0\) we have \(\beta =-\frac{\alpha \gamma }{\delta }\). Moreover the space is locally symmetric if either \(\alpha =\delta \) or \(\alpha =\gamma =0\). Now Eq. (12) becomes

$$\begin{aligned} 2\delta ^3\, \mathfrak {R}_{1213;1}= & {} (\alpha ^2-\delta ^2)(\gamma ^2+\delta ^2)((\alpha +\delta )\gamma +2\delta T_{123})=0,\\ \delta ^2\, \mathfrak {R}_{1213;2}= & {} (\alpha ^2-\delta ^2)(\gamma ^2+\delta ^2) T_{223}=0,\\ \delta ^2\, \mathfrak {R}_{1213;3}= & {} (\alpha ^2-\delta ^2)(\gamma ^2+\delta ^2) T_{323}=0,\\ -\delta ^2\, \mathfrak {R}_{1223;1}= & {} \alpha (\alpha -\delta )(\gamma ^2+\delta ^2) T_{113}=0,\\ 2\delta ^3\,\mathfrak {R}_{1223;2}= & {} \alpha (\alpha -\delta )(\gamma ^2+\delta ^2)((\alpha -\delta )\gamma -2\delta T_{213}) =0,\\ -\delta ^2\,\mathfrak {R}_{1223;3}= & {} \alpha (\alpha -\delta )(\gamma ^2+\delta ^2)(\delta +T_{313}) =0,\\ -\delta \,\mathfrak {R}_{1323;1}= & {} (\alpha -\delta )(\gamma ^2+\delta ^2)T_{112} =0,\\ -\delta \, \mathfrak {R}_{1323;2}= & {} (\alpha -\delta )(\gamma ^2+\delta ^2)(\alpha +T_{212}) =0,\\ 2\delta ^2\,\mathfrak {R}_{1323;3}= & {} (\alpha -\delta )(\gamma ^2+\delta ^2) ((\alpha -\delta )\gamma -2\delta T_{312})=0. \end{aligned}$$

Note that \(\alpha ^2-\delta ^2\ne 0\), since \(\alpha +\delta \ne 0\) and if \(\alpha -\delta =0\) then the space is locally symmetric. Moreover, if \(\alpha =0\) then \(\beta =0\) and the metric is isometric to the one in Sect. 3.1. Hence, the previous equations together with Eq. (10) imply that \(\widetilde{\nabla }_XY=0\) for left-invariant vector fields and the only homogeneous structure is the canonical one, which corresponds to

$$\begin{aligned} T= & {} -\tfrac{(\alpha +\delta )\gamma }{\delta } e^1\otimes (e^2\wedge e^3) -2\alpha e^2\otimes (e^1\wedge e^2) +\tfrac{(\alpha -\delta )\gamma }{\delta } e^2\otimes (e^1\wedge e^3)\\{} & {} + \tfrac{(\alpha -\delta )\gamma }{\delta } e^3\otimes (e^1\wedge e^2) -2\delta e^3\otimes (e^1\wedge e^3), \end{aligned}$$

and its projections are given by

$$\begin{aligned} p_1(T)= & {} -(\alpha +\delta ) \left( e^2\otimes (e^1\wedge e^2) + e^3\otimes (e^1\wedge e^3) \right) ,\\ p_2(T)= & {} -\frac{2(\alpha +\delta )\gamma }{3\delta } e^1\otimes (e^2\wedge e^3) -(\alpha -\delta ) e^2\otimes (e^1\wedge e^2) + \frac{2(\alpha -2\delta )\gamma }{3\delta } e^2\otimes (e^1\wedge e^3)\\{} & {} +\frac{2(2\alpha -\delta )\gamma }{3\delta } e^3\otimes (e^1\wedge e^2) +(\alpha -\delta ) e^3\otimes (e^1\wedge e^3),\\ p_3(T)= & {} -\frac{(\alpha +\delta )\gamma }{3\delta } \left( e^1\otimes (e^2\wedge e^3) - e^2\otimes (e^1\wedge e^3) + e^3\otimes (e^1\wedge e^2) \right) . \end{aligned}$$