Homogeneous Riemannian structures in dimension three

In this note, we determine all the homogeneous structures on non-symmetric three-dimensional Riemannian Lie groups. We show that a non-symmetric three-dimensional Riemannian Lie group admits a non-canonical homogeneous structure if and only if its isometry group has dimension four.

if there exists a (1, 2)-tensor field T on M such that where ∇ is the Ambrose-Singer connection given by ∇ = ∇ − T , ∇ 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 , 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 T (V) of such tensor fields on a vector space (V, , ) and decomposed it into three irreducible components under the action of the orthogonal group as T The subspaces of such decomposition are given as follows 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 T 3 and that a Riemannian manifold admits a non-vanishing structure of type 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 T 1 ⊕ T 3 , T / ∈ T 1 and T / ∈ 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 T 2 were described in [5] (see also [2]). Homogeneous structures in the class T 1 ⊕ 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 T 3 also admits a T 2 -structure.
In dimension two T (V) = T 1 (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 , is of special interest for our purposes. Let T ∇ be the canonical homogeneous structure defined by for left-invariant vector fields X , Y and Z . Then the corresponding Ambrose-Singer connection ∇ = ∇ − T ∇ satisfies ∇ X Y = 0 for left-invariant vector fields. This structure is equivalent to the description G = G/{e}, which corresponds to the action G × G → 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. 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 T 3 . In this case (M, g) is a real space form R 3 , S 3 or H 3 , or it is isometric either to the special unitary group SU (2), or to the universal cover of SL(2, 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 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 , , 1 ) is isometric to a second Lie group (G 2 , , 2 ) such that is not isomorphic to G 1 if and only if its isometry group has dimension at least 4.

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.

Unimodular Lie groups
Left-invariant Riemannian metrics , on unimodular Lie groups G were described by Milnor (see [7]) in terms of parameters (λ 1 , λ 2 , λ 3 ), so that the Lie algebra becomes (i) The three structure constants λ 1 , λ 2 , λ 3 are different and the only homogeneous structure is the canonical one, given by The canonical homogeneous structure is of type T 2 if λ 1 + λ 2 + λ 3 = 0 (see also [3]) and it is of type T 2 ⊕ T 3 otherwise. (ii) Up to a rotation, the structure constants λ 1 = λ 2 = λ 3 , λ 3 = 0 and there exists a one-parameter family of homogeneous structures which corresponds to the canonical structure for κ = 1 Unimodular Lie groups in Theorem 1.2-(ii) correspond to SU (2), SL(2, 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].

Non-unimodular Lie groups
Non-unimodular Riemannian Lie groups (G, , ) are semi-direct extensions R 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 In this case the homogeneous structure is of type which is of the generic type T 1 ⊕ T 2 ⊕ T 3 .
(ii) If δα = 0, β = − αγ δ and α = δ, then the only homogeneous structure is the canonical one, given by Non-unimodular Lie groups in Theorem 1.3 are semi-direct extensions R R 2 of the Abelian Lie group determined by an endomorphism − ad(e 1 ). Assertion (i) in Theorem 1.3 corresponds to the special situation det ad(e 1 ) = 0, and they are isometric (although not isomorphically isometric) to a left-invariant metric on SL(2, 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 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, 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 fourdimensional.
Remark 1.4 A more conceptual proof of this last statement can be summarized as follows. For any three-dimensional Lie groups (G 1 , , 1 ) and (G 2 , , 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 , , 1 ) and (G 2 , , 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 threedimensional isometry group is isometric to a unique Riemannian Lie group, in which case any homogeneous structure is isomorphic to the canonical one. 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 λ 1 = λ 2 = λ 3 = 0 (in which case the sectional curvature is constant and positive) or, up to a rotation, one has λ 1 = 0 and λ 2 = λ 3 (in which case the metric is flat).

Homogeneous structures on non-symmetric unimodular Lie groups
Let T be a (0, 3)-tensor field so that the connection ∇ = ∇ − T makes the metric tensor parallel, i.e., T xyz + T xzy = 0 for x, y, z ∈ 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 i j<k T i jk e i ⊗ (e j ∧ e k ).
Therefore, the non-zero components of the connection ∇ = ∇ − T are given by while the (0, 4)-curvature tensor field is determined by Let R ik j ;r = ( ∇ e r R)(e i , e j , e k , e ). A straightforward calculation using Eqs. (4) and (5) shows that the condition ∇ R = 0 in Eq. (1) is given by Next, depending on the eigenvalues λ i , we are led to the following two possibilities.

Case of two different eigenvalues
In this case, without loss of generality, we can assume λ 1 = λ 2 = λ 3 . Moreover, λ 3 = 0 since the space would be locally symmetric otherwise. Thus, Eq. (6) implies Let T i jk;r = ( ∇ e r T )(e i , e j , e k ). A straightforward calculation using Eqs. (4) and (9)  In the particular case where κ = 1 2 (2λ 1 −λ 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

Homogeneous structures on non-symmetric non-unimodular Lie groups
If g is non-unimodular then there exists an orthonormal basis {e 1 , e 2 , e 3 } of g such that (see [7]) where α + δ = 0 and αγ + βδ = 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 ad(e 1 ) is a multiple of the identity or it has complex eigenvalues), or it is locally isometric to a product R × N (c), where N (c) is a surface of constant negative sectional curvature (if ad(e 1 ) is of rank-one and {e 2 , e 3 } is an orthonormal basis of eigenvectors).