CLASSIFICATION OF LEFT INVARIANT HERMITIAN STRUCTURES ON 4-DIMENSIONAL NON-COMPACT RANK ONE SYMMETRIC SPACES

The only 4-dimensional non-compact rank one symmetric spaces are CH2 and RH4. By the classical results of Heintze, one can model these spaces by real solvable Lie groups with left invariant metrics. In this paper we classify all possible left invariant Hermitian structures on these Lie groups, i.e., left invariant Riemannian metrics and the corresponding Hermitian complex structures. We show that each metric from the classification on CH2 admits at least four Hermitian complex structures. One class of metrics on CH2 and all the metrics on RH4 admit 2-spheres of Hermitian complex structures. The standard metric of CH2 is the only Einstein metric from the classification, and also the only metric that admits Kähler structure, while on RH4 all the metrics are Einstein. Finally, we examine the geometry of these Lie groups: curvature properties, self-duality, and holonomy.

In dimension four this comes down to spaces CH 2 and RH 4 , since the quaternionic hyperbolic line HH 1 coincides with RH 4 . This follows from the isomorphism of Lie groups Sp(1, 1) ∼ = SO(1, 4)/±, giving HH 1 = Sp(1, 1)/Sp(1) × Sp(1) = SO(1, 4)/SO(4) = RH 4 . From Heintze's papers [9,10], we also know that a connected homogeneous manifold of non-positive curvature can be represented as a connected solvable Lie group with a left invariant metric 1 . RH 4 and CH 2 are both symmetric spaces of negative sectional curvature, therefore they are solvmanifolds, i.e., they can be represented as connected solvable Lie groups with left invariant metrics. They are exactly the noncompact part in the Iwasawa decomposition of the corresponding isometry groups. We denote these solvable Lie groups by RH 4 and CH 2 , and their Lie algebras by rh 4 and ch 2 . 2 In the sequel we classify the Hermitian structures on these groups: all possible non-isometric left invariant Riemannian metrics and Hermitian complex structures in regard to those metrics.
Left invariant Riemannian metrics. We give the classification of non-isometric left invariant Riemannian metrics on the Lie group CH 2 in Theorem 2.1. The classification of non-isometric left invariant Riemannian metrics on the Lie group RH 4 is presented in Theorem 3.1. These results are partially known (see [6, Table 2, A 1 4,9 for ch 2 , A 1,1 4,5 for rh 4 ]). In dimension three, Milnor [14] gave a complete solution to the more general problem: he used the cross product to classify all left invariant Riemannian metrics on 3-dimensional Lie groups. For the classification of metrics in higher dimensions we use different methods based on Alekseevsky's result that for completely solvable Lie groups the isometry classes of left invariant metrics are the orbits of the automorphism group acting on the space of left invariant metrics (see [1,2] and Lemmas 1.1, 1.2). From Jensen's classification of homogeneous Einstein spaces of dimension four [11], we know that CH 2 and RH 4 are the only non-compact, non-flat, indecomposable 4-dimensional Riemannian homogenous spaces that admit a left invariant Einstein metric. They are also found in [12] within a classification of 4-dimensional Einstein Lie groups. These results are in line with Alekseevsky's conjecture that every non-compact Riemannian Einstein homogenous space is a solvmanifold. A left invariant Riemannian Einstein metric on a given Lie group, if it exists, is unique up to a scaling (see [13] for recent results on Einstein solvmanifolds). Therefore, the Einstein metrics obtained on RH 4 and CH 2 in [11,12] are the usual metrics of hyperbolic spaces RH 4 and CH 2 viewed as symmetric spaces.
Hermitian complex structures. The main result of this paper is a classification of left invariant Hermitian complex structures on CH 2 with respect to all non-isometric left invariant Riemannian metrics (Theorem 2.2). Note that for each left invariant metric there exists a 2-dimensional sphere of Hermitian almost complex structures. However, the integrability condition imposes further restrictions, sometimes so strong that there are even-dimensional Lie groups that do not admit any integrable complex structures at all (see [8]). In our case, finding the integrable 1 Note that a simply connected, homogeneous manifold can be isometric to more than one Lie group equipped with a left invariant metric. In other words, non-isomorphic Lie groups might admit left invariant metrics which make them isometric as Riemannian manifolds. For a way to test whether two given solvmanifolds are isometric see [7]. 2 The Lie algebra ch 2 is denoted by d 4, 1 Hermitian structures on CH 2 amounts to intersecting the sphere of Hermitian almost complex structures with some non-trivial surfaces of second order coming from the Nijenhuis integrability condition. On RH 4 the integrability condition is automatically satisfied, so each metric allows a sphere of Hermitian complex structures (Theorem 3.2). The classification of left invariant Hermitian complex structures on CH 2 is related to many known results of complex geometry of Lie algebras. For example, ch 2 appears in Barberis's classification of invariant hypercomplex structures on 4-dimensional Lie groups [4]. These structures happen to be Hermitian, but not Kählerian (the first case from Table 1). In fact, it was shown in [5] that ch 2 is one of three non-abelian 4-dimensional Lie algebras admitting both the hypercomplex and the para-hypercomplex structure. Snow [18] investigated 4-dimensional solvable simply-connected real Lie groups with commutator subalgebra of dimension less than three. He classified left invariant complex structures with respect to certain subalgebras of the complexification of real Lie algebras. Ovando [16] extended this result to 4-dimensional Lie algebras with 3-dimensional commutator algebra. The ch 2 case (denoted by H 3 in [16]) is related to our result in the special case of the left invariant metric associated with an orthonormal base. Specially, Ovando [17] considered spaces CH 2 and RH 4 modelized as Lie groups and classified all left invariant complex structures with respect to standard metrics (see Remarks 2.2, 3.1). In the rh 4 case she found a sphere of Hermitian complex structures which is consistent with our result.
Outline. Section 1 explains the automorphism group action on the space of left invariant metrics. Basic notation and formulas are introduced here.
In Section 2 we classify Hermitian structures on CH 2 . The group of automorphisms of the Lie algebra ch 2 is given by Lemma 2.1. Theorem 2.1 classifies all nonisometric positive definite inner products on ch 2 . They are denoted by S(p, x, β), where p, β > 0, x ≥ 0 are real parameters, and p − x 2 > 0. The corresponding left invariant Riemannian metrics are denoted by g(p, x, β). Lemma 2.2 describes the isometric automorphisms of these metrics. In Theorem 2.2 we classify Hermitian complex structures on CH 2 with respect to g(p, x, β). Their matrix representations are listed in Table 1.
In Section 3 we classify Hermitian structures on RH 4 . Lemma 3.1 describes the automorphisms of the Lie algebra rh 4 , and Theorem 3.1 classifies non-isometric positive definite inner products S(λ). We prove that each of the metrics g(λ), corresponding to S(λ), has a 2-dimensional sphere of Hermitian complex structures (Theorem 3.2).
Section 4 considers geometry of the metric Lie groups CH 2 and RH 4 . Curvature properties of CH 2 are given in Theorem 4.1. In Theorem 4.2 we show that all metrics g(p, x, β) have the full holonomy, except the Kähler metric g( 1 β , 0, β) whose holonomy group is U (2). Theorem 4.3 shows that all metrics g(λ) on RH 4 have constant negative sectional curvature and full holonomy group.
We are very grateful to our colleagues Neda Bokan and TijanaŠukilović for many valuable comments and suggestions.

Preliminaries
A Lie algebra g together with an inner product is called a metric Lie algebra. Every metric Lie algebra defines, by left translations, the unique left invariant metric on the corresponding Lie group G, called a metric Lie group. Two metric Lie algebras are isometric if they are isomorphic as vector spaces, and the isomorphism between them preserves curvature tensor and its covariant derivatives. In other words: Lemma 1.1 (Alekseevsky [2]). Two metric Lie algebras are isometric if and only if the corresponding Riemannian spaces are isometric.
In general, two isometric metric Lie algebras do not have to be isomorphic. However, this does hold for completely solvable Lie algebras.

Lemma 1.2 (Alekseevsky [1, 2]). Isometric completely solvable metric Lie algebras are isomorphic.
Therefore, the isometry classes of different left invariant metrics are precisely the orbits of the automorphism group Aut(g) acting on the space of left invariant metrics. Since CH 2 and RH 4 are completely solvable groups, this allows us to classify non-isometric metrics using algebraic methods in Theorems 2.1 and 3.1.
Recall that an almost complex structure on a manifold M is an automorphism The space of Hermitian orientation preserving almost complex structures on 2n-dimensional vector space with a given positive definite inner product is the symmetric space SO(2n)/U (n). For 2n = 4 we have SO(4)/U (2) = S 2 , the two dimensional sphere of the almost complex Hermitian structures.
The Nijenhuis tensor N of an almost complex structure J is a tensor field of rank (1,2) given by for any X, Y ∈ T M . If an almost complex structure J is integrable, i.e., its Nijenhuis tensor vanishes identically, then J is a complex structure and M is a complex manifold. In Theorems 2.2 and 3.2 we essentially look for those almost complex Hermitian structures that annihilate the Nijenhuis tensor.
Note that a Lie group admitting a left invariant complex structure may not be a complex Lie group. In order to have the structure of a complex Lie group, the group multiplication has to be holomorphic, or equivalently, the complex structure J has to be ad-invariant.
Let (M, g) be a Riemannian manifold and ∇ its Levi-Civita connection. When the manifold M is a Lie group, Koszul's formula reduces to for any left invariant vector fields X, Y, Z. We define the Riemann curvature operator R(X, Y ) : for any vector fields X, Y on T p M . The Ricci curvature tensor is defined by A manifold (M, g) is Einstein if ρ is proportional to the metric g. The scalar curvature τ is the trace of the Ricci tensor with respect to the metric g.
is a sectional curvature in the plane π. If M is endowed with a Hermitian complex structure J, then for a unit vector X ∈ T p M , vectors X and JX form an orthonormal basis of the J-invariant plane π and K(π) is called a holomorphical sectional curvature.
and therefore belong to the algebra of skew-symmetric endomorphisms so(g). We identify the skew-symmetric endomorphisms so(g) , the curvature tensor can be regarded as an operator on the space of 2-vectors A Riemannian metric on T p M naturally induces a positive definite inner product ·, · on the space of k-vectors Λ k T p M by It is symmetric with respect to the inner product ·, · and satisfies * 2 = Id. Therefore, it induces an orthogonal splitting With respect to that splitting, the curvature tensor has the well known decomposition where τ denotes the scalar curvature, W ± is the Weyl conformal tensor, and B is the traceless Ricci tensor. We say that Consider the Iwasawa decomposition, SU (1, 2) = KAN , of the semisimple group of isometries of CH 2 . The compact part is isomorphic to U (2), the nilpotent part N is the Heisenberg group H 3 , and the abelian part is 1-dimensional. The completely solvable group CH 2 = AN acts simply transitively on the complex hyperbolic plane CH 2 and gives it a structure of a Lie group in a way that its Kähler metric is left invariant. Its Lie algebra ch 2 is spanned by vectors X, Y, Z, W with nonzero commutators: ch 2 is a semidirect product of the nilpotent Heisenberg ideal spanned by Y, Z, W and the 1-dimensional center spanned by X.
CH 2 is a solvmanifold in a trivial way, i.e., it acts on itself isometrically by left translations. Equipped with the Kähler left invariant metric, CH 2 is the complex hyperbolic plane CH 2 in the usual sense.
Theorem 2.1. All non-isometric positive definite inner products on ch 2 , in some basis with Lie algebra commutators (10), are represented by the matrices where Proof. Two different inner products on ch 2 define two metric Lie algebras on the same space. If the inner products are represented by symmetric positive definite matrices S and F T SF, F ∈ Aut(ch 2 ), those metric Lie algebras are isomorphic, and therefore they are isometric. The converse statement follows from Lemma 1.2. Thus, non-isometric left invariant inner products correspond to the orbits of the group Aut(ch 2 ) acting on the space of positive definite symmetric matrices. Lengthy calculations similar to the proof of Theorem 3.1 (rh 4 case) show that these orbits are represented by the matrices of the form S(p, x, β).
A left invariant metric on a Lie group is uniquely determined by an inner product on its Lie algebra. We denote by g(p, x, β) the left invariant Riemannian metric on CH 2 corresponding to the inner product S (p, x, β). From Lemma 1.1 it follows that the isometry classes of the metrics on CH 2 are the orbits of the automorphism group Aut(CH 2 ) acting on the space of left invariant metrics. Therefore, Theorem 2.1 gives us the classification of all non-isometric left invariant metrics on CH 2 . Proof. The matrix of an automorphism F is of the form (11). It is an isometry if F T SF = S is satisfied.

Matrices of these structures and their isometric isomorphism classes are given in
Proof. Let J = (a ij ) denote the matrix representing the complex structure J in a basis with commutators (10). Since J is a Hermitian complex structure, it imposes certain constraints on the entries of the matrix J = (a ij ).
The condition (2) that the structure J is Hermitian, is equivalent to

S(p, x, β)
Hermitian complex structures Isomorphic  The condition that J is a complex structure is equivalent to the system of quadratic equations where the Nijenhuis tensor N is given by (3). These quadratic constraints lead us to discuss the following cases: Conditions (13) are now equivalent to 4a 2 31 β + 4a 2 41 β 2 + a 2 43 β = 1. Denoting r = 2a 31 β, s = a 43 β, t = 2a 41 β, we have the sphere of complex structures J r,s,t , with r 2 + s 2 + t 2 = 1, listed in Table 1.
Conditions (13) Consequently, in this case we have four complex structures J K± , J N K± , listed in Table 1.
From conditions (13) and the regularity of J it follows that a 43 = 0, a 12 = 0, From conditions (15) and (16) (a 41 , a 42 ). Since p = 0, x = 0, the equation (17) is always a hyperbola, and the equation (18) is an ellipse. One can check that these curves always intersect in 4 points given by All four possible values of a 41 are real under the given conditions. Denoting we have matrices of the corresponding complex structures J 1± and J 2± given in Table 1.
From conditions (15) and (16) and the regularity of the matrix J it follows that Conditions (13) are now equivalent to The equation (19) with variables a 32 and a 41 represents a hyperbola. Notice that the determinant of the quadratic form (20) equals −(1 − 4pβ) 2 det S. It is strictly negative, so the equation (20) represents two lines intersecting at the origin. Therefore, the solution of the system above is the intersection of a hyperbola and two lines. The coefficient of the term a 2 32 in (20) equals 1 β (4p 2 β 2 − det S) which leads us to consider the following cases: In this case, a 41 = 0. We put a 32 = ta 41 and from (20) we get The corresponding complex structures J 3± and J 4± are given in Table 1. We use the notation t 3 = t + , t 4 = t − , q = a 41 . The only two options, a 41 = 0 and a 41 = 0, give structures J 5± and J 6± respectively. This covers all the cases, so all complex structures have been found. If two structures J and F −1 JF are related by an isometric automorphism from Lemma 2.2 they determine the same geometry. We call them isomorphic and they are listed in Table 1.

Remark 2.2.
Ovando [16,17] found three classes of left invariant complex structures with respect to certain subalgebras of the complexification of the real Lie algebra ch 2 . Considering an orthonormal basis (thus fixing one left invariant metric) she noted that one of those classes contains the standard Kähler metric on CH 2 . This case corresponds to J K± from our classification in the special case of the left invariant metric g (1, 0, 1). The second class from [16] corresponds to J N K± with the same metrics, and complex structures from the third class are not Hermitian.

Classification of left invariant Hermitian structures on RH 4
If SO(1, 4) = KAN is the Iwasawa decomposition of the group of isometries of RH 4 , then the compact part is SO(4), the abelian part is R 3 , and the nilpotent part is 1-dimensional. The Lie group RH 4 = AN , acting simply transitively on the real hyperbolic space RH 4 , has the Lie algebra rh 4 spanned by vectors e 1 , e 2 , e 3 , e 4 with nonzero commutators: It is a two step nilpotent algebra with a 3-dimensional commutator subalgebra spanned by e 1 , e 2 , e 3 .

Lemma 3.1. The automorphism group of rh 4 is
This group is isomorphic to the group Aff 3 (R) of affine transformations of R 3 .
Proof. Direct calculation shows that conditions of integrability (Nijenhuis) are a consequence of conditions J 2 = −Id and J being Hermitian. Now we are ready to prove the theorems that describe curvature properties of left invariant metrics g(p, x, β) on CH 2 , and of left invariant metrics g(λ) on RH 4 . i) For all left invariant Riemannian metrics g(p, x, β), CH 2 has a constant negative scalar curvature ii) Among g(p, x, β), the unique Einstein metric is g( 1 β , 0, β), β > 0. Moreover, it is the unique metric that admits a Kähler structure (J K± in classification). CH 2 with this metric is a solvmanifold with constant holomorphic sectional curvature −β and it is isometric to the complex hyperbolic plane CH 2 . iii) g( 1 4β , 0, β) and g( 1 β , 0, β) are the only half-conformally flat metrics on CH 2 .
Proof. Since we work with left invariant basis, we have a natural identification T p CH 2 ∼ = ch 2 for any p ∈ CH 2 . Let (X, Y, Z, W ) be a basis of ch 2 with commutators of the form (10). In that basis, the metric g(p, x, β) is represented by the matrix S = S(p, x, β) of the form (12). The Levi-Civita connection can be calculated by Koszul's formula (4). Using the fact that ∇ is torsion-free, ∇ X1 X 2 − ∇ X2 X 1 = [X 1 , X 2 ], it follows that all nonzero Therefore, the only Einstein left invariant metric on CH 2 is g( 1 β , 0, β). For the complex structure J K± from Table 1, direct calculation shows ∇ V J K± = 0 for any V ∈ ch 2 . Thus, (CH 2 , g( 1 β , 0, β), J K± ) is a Kähler manifold with constant holomorphic sectional curvature −β, so it is isometric to the complex hyperbolic plane CH 2 . Note that g( 1 β , 0, β) with complex structure J N K± is not Kählerian, but also has a constant holomorphic sectional curvature −β.
We check case-by-case structures from Table 1 to prove that no metric other than g( 1 β , 0, β) admits a Kählerian structure. This proves part ii). To prove iii) we introduce an orthonormal basis of ch 2 with respect to the inner product S(p, x, β) The basis e i ∧ e j , i < j, is an orthonormal basis of Λ 2 ch 2 , and ω = e 1 ∧ e 2 ∧ e 3 ∧ e 4 is a unit vector that spans Λ 4 ch 2 . Further lengthy calculations show that (26) are orthonormal bases of ±1 eigenspaces Λ 2 ± of the Hodge operator * defined by (8). One can use R : Λ 2 → Λ 2 given by (24) with respect to the basis (26) to obtain the decomposition (9). If x = 0 then W ± = 0, so a metric cannot be half-conformally flat. If x = 0 we obtain We see that W − vanishes when pβ = 1 or 4pβ = 1, so iii) is proved.  , 0, β), whose holonomy group is isomorphic to U (2).
Proof. The infinitesimal holonomy algebra is spanned by curvature operators R(·, ·) and their covariant derivatives ∇ k R of all degrees k ∈ N. The corresponding uniquely determined connected subgroup of GL(4, R) is called the infinitesimal holonomy group. Since a Lie group is an analytic manifold, its restricted, local and infinitesimal holonomy groups coincide (see [15]). Moreover, CH 2 is simply connected, therefore, the holonomy group is the same as the local holonomy group.
In the pβ = 1, x = 0 case the metric is symmetric. Hence, ∇R = 0 and the holonomy algebra is u(2) spanned only by curvature operators. In all other cases, direct but lengthy calculations show that the holonomy algebra is isomorphic to so (4) and is spanned by curvature operators and their first covariant derivatives.
We may now conclude that in the pβ = 1, x = 0 case the holonomy group is U (2). In all other cases, the metric g has the full holonomy group.  Further calculation brings us a nice expression for the curvature operator R : Λ 2 rh 4 → Λ 2 rh 4 given on the basis of the space of 2-vectors Λ 2 rh 4 : From definition (7) and the previous formula, it follows that the sectional curvature is K(π) = − 1 λ , for any 2-dimensional plane π ⊂ T p RH 4 . Using (6) we find the matrix of the Ricci tensor: ρ = diag(− 3 λ , − 3 λ , − 3 λ , −3). Since ρ is proportional to the inner product matrix (23), the corresponding left invariant metric g(λ) is Einstein, and the scalar curvature is τ = − 12 λ . The holonomy algebra is 6-dimensional, hence g(λ) has the full holonomy group.