Four-Dimensional Pseudo-Riemannian Generalized Symmetric Spaces Which are Algebraic Ricci Solitons

We classify, up to isometry, non-symmetric simply-connected four-dimensional pseudo-Riemannian generalized symmetric spaces which are algebraic Ricci solitons. It turns out that those of Cerny–Kowalski’s types A, C and D are algebraic Ricci solitons, whereas those of type B are not. Thus, we give new examples of algebraic Ricci solitons. Mathematics Subject Classification (2000). 53C50, 53C21, 53C25.


Introduction
The concept of algebraic Ricci soliton was introduced by Lauret in the Riemannian case ( [9]). The definition extends to the pseudo-Riemannian case as follows.
Definition 1.1. Let (G, g) be a simply-connected Lie group equipped with a left-invariant pseudo-Riemannian metric g, and let g denote the Lie algebra of G. The metric g is called an algebraic Ricci soliton if it satisfies Ric = c Id + D, (1.1) where L denotes the Lie derivative, is the Ricci tensor and c is a real constant. The condition (1.3) is equivalent to g t = (−2ct+1)ϕ * s(t) g being a solution of the Ricci flow ∂ ∂t where ϕ s is the family of diffeomorphisms generated by X which one reparametrizes to s(t) = 1 c ln(−2ct + 1). In [9], Lauret studied the relation between solvsolitons and Ricci solitons on Riemannian manifolds. More precisely, he proved that any left-invariant Riemannian solvsoliton metric is a Ricci soliton. This was extended by the second author to the pseudo-Riemannian case as follows. where e denotes the identity element of G.
Note that changing "solvsoliton" to "algebraic Ricci soliton" the theorem above is correct.
On the other hand, if (M, g) is a homogeneous (pseudo-)Riemannian manifold, there exists a group G of isometries acting transitively on it [11]. Such (M, g) can be then identified with (G/H, g), where H is the isotropy group at a fixed point p of M . Let g denote the Lie algebra of G and fix an Ad(H)-invariant decomposition g = h ⊕ m, where h is the Lie algebra of H. The space m is naturally identified with T p M . In the Riemannian case such a decomposition always exists, since homogeneous Riemannian manifolds are reductive. In the general pseudo-Riemannian, reductivity should generally instead be imposed. Now, for instance, a three-dimensional homogeneous Lorentzian manifold is necessarily reductive. This was proved in [7] and it also follows independently from the classification obtained by Calvaruso in [2]. Furthermore, the existence of non-reductive four-dimensional pseudo-Riemannian homogeneous manifolds was proven in [7].
Homogeneous Ricci solitons have been investigated in [8]. A natural generalization of algebraic Ricci solitons on Lie groups to homogeneous spaces is the following [8]. where Ric denotes the Ricci operator on m, pr: g → m is the orthogonal projection map, c is a real number, and D ∈ Der(g).
Note that the above definition can be extended to the pseudo-Riemannian case, changing "homogeneous Riemannian manifold" to "reductive homogeneous pseudo-Riemannian manifold". In [1], we obtained the classification of three-dimensional Lorentzian Lie groups which are algebraic Ricci solitons.
In [4], pseudo-Riemannian four-dimensional generalized symmetric spaces have been classified into four classes, named A, B, C and D, and the (pseudo-) Riemannian metrics can have any signature. All these spaces are reductive homogeneous.
In [3] and [6], the Levi-Civita connection, the curvature tensor and the Ricci tensor of these spaces are computed; proving that type C is symmetric, that is, the covariant derivative of its curvature tensor vanishes at each point. We will use the results of these computations to study which types of these spaces are algebraic Ricci solitons.
The main result of this paper can be stated (cf. Theorems 3.3, 4.3 , 5.3, 6.3) as follows.

Preliminaries
We start by recalling the definition of generalized symmetric space. Let (M, g) be a (pseudo-)Riemannian manifold. A regular s-structure on M is a family of isometries The map s p is called the symmetry centered at p. The order of a regular s-structure is the smallest integer k 2 such that s k p = id M for all p ∈ M . If such an integer does not exist, we say that the regular s-structure has order 256 W. Batat and K. Onda Results. Math. infinity. A generalized symmetric space is a connected, pseudo-Riemannian manifold, carrying at least one regular s-structure. In particular, a generalized symmetric space is a pseudo-Riemannian symmetric space if and only if it admits a regular s-structure of order 2. The order of a generalized symmetric space is the minimum of orders of all possible s-structures on it. Furthermore, if (M, g) is a generalized symmetric space then it is homogeneous, that is, the full isometry group I(M ) of M acts transitively on it, which means that (M, g) can be identified with (G/H, g), where G ⊂ I(M ) is a subgroup of I(M ) acting transitively on M and H is the isotropy group at a fixed point o ∈ M .
Generalized symmetric spaces of low dimension have been completely classified. The following theorem recalls the classification of non-symmetric simply-connected four-dimensional pseudo-Riemannian generalized symmetric spaces. [4]) Non-symmetric, simply-connected generalized symmetric spaces (M, g) of dimension 4 are of order either 3 or 4, or infinity. All these spaces are indecomposable, and belong, up to isometry, to one of the following four types.
where λ is a real constant. The order is k = 3 and the signature is always (2, 2).

4)
where λ = 0 is a real constant. The order is infinite and the signature is (2, 2).

Spaces of Type A with Neutral Signature
Let (M, g) be a four-dimensional generalized pseudo-Riemannian symmetric space and denote by ∇ and R the Levi-Civita connection and the Riemann curvature tensor of M , respectively. Throughout this paper, we will use the sign convention The Ricci tensor of (M, g) is defined by where {e 1 , e 2 , e 3 , e 4 } is a pseudo-orthonormal frame field, with g(e k , e k ) = ε k = ±1. The Ricci operator Ric is then given by (X, Y ) = g(Ric(X), Y ). Now, consider a non-symmetric simply-connected four-dimensional generalized symmetric space (M = G/H, g) of type A and signature (2,2). Then, taking into account the results of [4] and [5], the Lie algebra g of the Lie group G may be decomposed into the vector space direct sum g = h ⊕ m where h denotes the Lie algebra of H and m is a vector subspace of g.
The Lie algebra g admits a basis {U 1 , U 2 , U 3 , U 4 , U 5 }, where {U 1 , U 2 , U 3 , U 4 } is an orthogonal basis of m and {U 5 } a basis of h, such that the Lie bracket [ , ] on g and the scalar product , on m are given by where δ > 0 is a real constant, and respectively. We now recall the following result on the curvature tensor and the Ricci tensor of four-dimensional generalized symmetric spaces of type A (see [3]).

The only non-zero components of the
A standard computation proves that all solutions of (3.2) are given by 3 5 . So, we have proved the following Using the lemma above, we now prove the following.

Spaces of Type B
Let (M = G/H, g) be a non-symmetric simply-connected four-dimensional generalized symmetric space of type B and signature (2,2). Then, g = h ⊕ m and {U 1 , U 2 , U 3 , U 4 } and {U 5 } are bases of m and h, respectively, such that the Lie bracket [ , ] on g and the scalar product , on m are given by where ε = ±1, and The following result was proven in [3].

Lemma 4.1.
Let M be a four-dimensional generalized symmetric space of type B and signature (2,2). There exists a pseudo-orthonormal frame field The only non-zero components of the Riemann curvature tensor R, with respect to {e 1 , e 2 , e 3 , e 4 }, are and the non-zero components of the Ricci tensor are given by 11 = 22 = 33 = 44 = −2, 13 = 24 = −4. Next, let D ∈ Der(g), where g is the Lie algebra in (4.1) and put l U 5 for all l = 1, . . . , 5. Using (4.1) , we prove that (1.2) is satisfied if and only if . So, we need to consider two cases: • If ε = 1, it is easily seen that all solutions of (4.2) are given by all solutions of (4.2) are given by . Therefore, we have proved the following Results. Math.
• ε = −1: Using the lemma above, we now prove the following. Proof. Using Lemma 4.1 we obtain that the Ricci operator of (M = G/H, g) is given, with respect to the basis {U 1 , U 2 , U 3 , U 4 , U 5 }, by Hence it follows, from the lemma above, that the algebraic Ricci soliton condition (1.4) on M is not satisfied.

Spaces of Type C
Let (M = G, g) be a non-symmetric simply-connected four-dimensional symmetric space of type C. Without loss of generality, we assume that the signature is (3, 1). The Lie algebra g admits a basis {U 1 , U 2 , U 3 , U 4 }, such that the Lie bracket [ , ] and the scalar product , on g are given by respectively.
The following result was proven in [6]. We can now prove the following.