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Two-symmetric Lorentzian manifolds

We classify two-symmetric Lorentzian manifolds using methods of the theory of holonomy groups. These manifolds are exhausted by a special type of pp-waves and, like the symmetric Cahen-Wallach spaces, they have commutative holonomy.


Introduction
Symmetric pseudo-Riemannian manifolds constitute an important class of spaces. A direct generalization of these manifolds is provided by the so-called k-symmetric pseudo-Riemannian spaces (M, g) satisfying the condition ∇ k R = 0, ∇ k−1 R = 0, where k ≥ 1 and R is the curvature tensor of (M, g). For Riemannian manifolds, the condition ∇ k R = 0 implies ∇R = 0 [17]. On the other hand, there exist pseudo-Riemannian k-symmetric spaces with k ≥ 2, see e.g. [13,15,1]. The fundamental paper by J.M. Senovilla [15] is devoted to a detailed investigation of two-symmetric Lorentzian spaces. It contains many interesting results about such manifolds and their physical applications. In particular, it is proven there that any two-symmetric Lorentzian space admits a parallel null vector field. A classification of four-dimensional two-symmetric Lorentzian spaces is obtained in the paper [1], in which it is shown that these spaces are some special pp-waves. The result is based on the Petrov classification of the Weyl tensors.
In the present paper we generalize the result of [1] to any dimension. The main result can be stated as follows.
where H ij is a nonzero diagonal real matrix with the diagonal elements λ 1 ≤ · · · ≤ λ n , and F ij is a symmetric real matrix.
Any other metric of this form isometric to g is given by the same H ij and byF ij = cH ij + F kl a k i a l j , where c ∈ R and a j i is an orthogonal matrix such that H kl a k i a l j = H ij .
By the Wu Theorem [19], any Lorentzian manifold (M, g) is either locally indecomposable, or it is locally a product of a Riemannian manifold (M 1 , g 1 ), and of a locally indecomposable Lorentzian manifold (M 2 , g 2 ). The manifold (M, g) is two-symmetric if and only if (M 1 , g 1 ) is locally symmetric and (M 2 , g 2 ) is two-symmetric. Consequently, Theorem 1 provides the complete local classification of two-symmetric Lorentzian manifolds.
For the proof of Theorem 1, we use the methods of the theory of holonomy groups. The assumption that a Lorentzian manifold (M, g) is two-symmetric implies that the holonomy algebra hol m of (M, g) at a point m ∈ M annihilates the tensor ∇R m = 0. This cannot happen if the holonomy algebra is the whole Lorentzian algebra so(1, n + 1). Hence the holonomy algebra must preserve a null line and is contained in the similitude algebra, hol m ⊂ sim(n) = (R ⊕ so(n)) + R n , the maximal Lie algebra with this property [6]. It is sufficient to consider the following two cases: hol m = h + R n , where h ⊂ so(n) is an irreducible subalgebra, and hol m = R n .
We prove that the first case is impossible: for this we calculate ∇R and ∇ Ric, and show that the Weyl conformal tensor W is parallel (∇W = 0). Then, using the results of A. Derdzinski and W. Roter [4,5] and of [10], we get a contradiction.
The second case corresponds to pp-waves. The condition ∇ 2 R = 0 and simple computations allow us to find the coordinate form of the metric.

Holonomy groups of Lorentzian manifolds
We recall some basic facts about holonomy groups of Lorentzian manifolds that can be found in [6,8,14]. Let (M, g) be a Lorentzian d-dimensional manifold and Hol 0 (M ) = Hol 0 (M ) m its connected holonomy group at a point m ∈ M . It is a subgroup of the (connected) Lorentz group SO(V ) 0 where V = T m M is the tangent space and it is determined by its Lie algebra hol(M ) ⊂ so(V ) which is called the holonomy algebra of M .
The manifold (M, g) is locally indecomposable (i.e. locally is not a direct product of two pseudo-Riemannian manifolds) if and only if the holonomy group Hol 0 (M ) (or the holonomy algebra hol(M )) is weakly irreducible, i.e. it does not preserve any proper nondegenerate subspace of V . Any weakly irreducible holonomy group Hol(M ) different from the Lorentz group SO(V ) 0 is a subgroup of the horospheric group SO(V ) [p] , the subgroup of SO 0 (V ) which preserves a null line [p] = Rp. This group is identified with the group Sim n = R * · SO n · R n , n = d − 2 of similarity transformations of the Euclidean space E = R n as follows (see [6,Sect. 2.3]). The Lorentzian group SO(V ) 0 acts transitively on the celestial sphere S n = P V 0 (the space of null lines) which is the projectivization of the null cone V 0 ⊂ V with the stabilizer SO(V ) [p] . The stabilizer has an open orbit S n \ [p] which is identified via the stereographic projection with the Euclidean space E. The group SO(V ) [p] acts in E as the full connected Lie group of similarity transformations. Having in mind this isomorphism, we will call the group SO(V ) [p] the similarity group and denote it by Sim n .
Using the metric < ., . >= g m , we will identify the Lorentz Lie algebra so(V ) ≃ so(1, n + 1) with the space Λ 2 V of bivectors. Then the Lie algebra sim n of the similarity group can be written as where p, q are isotropic vectors with < p, q >= 1 which span 2-dimensional Minkowski subspace U and E = U ⊥ is its orthogonal complement. The commutative ideal p ∧ E generates the commutative normal subgroup T E ⊂ Sim n which acts on E by translations. This group is called the vector group. The onedimensional subalgebra Rp ∧ q = so(U ) generates the maximal diagonal subgroup A of Sim n which is the Lorentz group SO(U ) 0 and the maximal compact subalgebra so(E) generates the group SO(E) of orthogonal transformations of E. The above decomposition of the Lie algebra sim n defines the Iwasawa decomposition of the group Sim n . The list of connected weakly irreducible holonomy groups Hol 0 (M ) of Lorentzian manifolds is known, see [14,6]. Assume for simplicity that Hol 0 (M ) is an algebraic group. Then it contains the vector group T E and has one of the following forms: is a connected holonomy group of a Riemannian ndimensional manifold, i.e. a product of the Lie groups from the Berger list: SO k , U k , SU k , Sp 1 · Sp k , Sp k , G 2 , Spin 7 and the isotropy groups of irreducible symmetric Riemannian manifolds.
If the holonomy group is not algebraic, it is obtained from one of the holonomy groups of type I or II by some twisting (holonomy groups of type III and IV). Note that all these holonomy groups act transitively on the Euclidean space E = P V 0 \ [p] [8].
The Lorentzian holonomy algebras g ⊂ sim(n) are the following : A locally indecomposable simply connected Lorentzian manifold admits a parallel null vector field if and only if its holonomy group is of type II or IV.
3. The holonomy group of a two-symmetric Lorentzian manifold So, one-symmetric spaces are the same as nonflat locally symmetric spaces (∇R = 0, R = 0). Recall that a complete simply connected locally symmetric space is a symmetric space, that is it admits a central symmetry S m with center at any point m, i.e. an involutive isometry S m which has m as an isolated fixed point.
Remark that for a Riemannian manifold the condition ∇ k R = 0 implies ∇R = 0 [17]. All indecomposable simply connected Lorentzian symmetric spaces are exhausted by the De Sitter and the anti De Sitter spaces and the Cahen-Wallach spaces, which have the vector holonomy group T E .
The following result is proven, using so-called casual tensors and the super-energy techniques, in [15].

Theorem 2. [15] Any two-symmetric Lorentzian manifold (M, g) admits a parallel null vector field.
This implies that the holonomy group can be only of type II or IV. To make the exposition complete, we will sketch a proof of Theorem 2 using the holonomy theory.
The corner stone of the paper is the following statement.
It is known that any (n + 2)-dimensional Lorentzian manifold with the holonomy algebra p ∧ E is a pp-wave (see e.g. [6, Sect 5.4]), i.e. locally there exist coordinates v, x 1 , ..., x n , u such that the metric g can be written in the form We will need only to decide which functions H correspond to two-symmetric spaces.

Algebraic curvature tensors
For a subalgebra g ⊂ so(V ) define the space of algebraic curvature tensors of type g, If g ⊂ so(V ) is the holonomy algebra of a manifold (M, g), where V = T m M is tangent space at some point m ∈ M , then the curvature tensor R m of (M, g) belongs to R(g). The spaces R(g) for holonomy algebras of Lorentzian manifolds are found in [7,9]. For example, let g = Rp ∧ q + h + p ∧ E. For a subalgebra h ⊂ so(n) define the space Any R ∈ R(g) is uniquely determined by the data (λ, e, P, R 0 , T ), where i.e. T is a symmetric tensor considered as an endomorphism of E. The tensor R is defined by We will write R = R (λ,e,P,R 0 ,T ) .
If some of these elements are zero, we omit them. For example, if R is defined only by T , then we write R = R T . Note that where e 1 , ..., e n is an orthonormal basis of E, and ∨ denotes the symmetric product. Similarly, Now we define the space of covariant derivatives of the curvature tensor If g ⊂ so(V ) is the holonomy algebra of a manifold (M, g) at a point m ∈ M , then ∇R m ∈ ∇R(g). The decomposition of the space ∇R(so(r, s)) into irreducible so(r, s)-modules is found in [16], see also [12]. It is not difficult to find the space ∇R(g) for each Lorentzian holonomy algebra g ⊂ sim(n). It consists of tensors S ∈ Hom(V, R(g)), S : u ∈ V → S u = R (λu,eu,Pu,R 0 u ,Tu) ∈ R(g) satisfying the second Bianchi identity. For example,
Let (M, g) be a Lorentzian manifold with the holonomy algebra g ⊂ sim(n). Then (M, g) admits a parallel distribution of null lines. According to [18], locally there exist so called Walker coordinates v, x 1 , ..., x n , u such that the metric g has the form where h = h ij (x 1 , ..., x n , u)dx i dx j is an u-dependent family of Riemannian metrics, A = A i (x 1 , . . . , x n , u)dx i is an u-dependent family of one-forms, and H = H(v, x 1 , ..., x n , u) is a local function on M . Consider the local frame Let E be the distribution generated by the vector fields X 1 ,...,X n . Clearly, the vector fields p, q are isotropic, g(p, q) = 1, the restriction of g to E is positive definite, and E is orthogonal to p and q. The vector field p defines the parallel distribution of null lines and it is recurrent, i.e. ∇p = θ ⊗ p, where θ = 1 2 ∂ v Hdu. Since the manifold is locally indecomposable, any other recurrent vector field is proportional to p. Next, p is proportional to a parallel vector field if and only if dθ = 0, which is equivalent to In the last case the coordinates can be chosen in such a way that ∂ v H = 0 and ∇p = ∇∂ v = 0, see e.g. [6].
Let g ⊂ sim(n) be the holonomy algebra of the Lorentzian manifold (M, g) and h ⊂ so(E) be its orthogonal part. Then there exist the decompositions such that h annihilates E 0 , h i (E j ) = 0 for i = j, and h i ⊂ so(E i ) is an irreducible subalgebra for 1 ≤ i ≤ s. Ch. Boubel [2] proved that there exist Walker coordinates v, x 0 = (x 1 0 , . . . , x n0 0 ), . . . , x r = (x 1 r , ..., x nr r ), u adapted to the decomposition (3.2). This means that The coordinates can be chosen so that A = 0, see [11]. Thus we will assume that g is given by (3.1) with A = 0, and with h satisfying (3.3) and (3.4). For α = 0, ..., r, consider the submanifolds M α ⊂ M defined by x β = c β , α = β, where c β are constant vectors. Then the induced metric is given by Lemma 1. The submanifold M α ⊂ M is totally geodesic. The orthogonal part of the holonomy algebra g α of the metrics g α coincides with h α ⊂ so(E α ), which is irreducible for α = 1, ..., r. If the metric g is two-symmetric, then the curvature tensor of each metric g α satisfies ∇ 2 R = 0.
Proof. The non-zero Christoffel symbols of the metric (3.1) with A = 0 are the following: where the comma denotes the partial derivative and Γ i jk (h) are the Christoffel symbols of the metric h. This shows that the Christoffel symbols of the metric g α are equal to the restrictions to M α of the corresponding Christoffel symbols of the metric g, i.e. each submanifold M α ⊂ M is totally geodesic. This implies that if ∇ 2 R = 0, then each g α satisfies the same condition. Finally, the statement about the orthogonal parts follows from the fact that the orthogonal part of any Walker metric g coincides with the holonomy algebra of the induced connection on the vector bundle with the fibers p ⊥ m /Rp m ≃ E m , and this connection does not depend on the function H [6].

Sketch of the proof of Theorem 2 using the holonomy theory
We may assume that the metric g is locally given by (3.1) with A = 0, and with h satisfying (3.3) and (3.4). As it is noted above, it is enough to prove that Clearly, this will be true if it is true for each metric g α .

Lemma 2.
If g is the holonomy algebra of type I (with any orthogonal part h ⊂ so(E)), or g is the holonomy algebra of type III with an irreducible orthogonal part h ⊂ so(E), then the subspace ∇R(g) 0 ⊂ ∇R(g), consisting of tensors annihilated by g, is trivial.
Proof. If g is of type I, then it contains A = p ∧ q. If g is of type III, then h ⊂ u(E) ⊂ so(E) and for some a ∈ R, the element A = p ∧ q + aJ belongs to g. The lemma follows from the consideration of the tensors in ∇R(g) annihilated by the operator A and the second Bianchi identity as in Lemma 3 below.
The lemma shows that the holonomy algebra of each metric g α cannot be of type I or III, i.e. it is of type II or IV. Thus,

Proof of Theorem 3
Consider the decomposition (3.2). If E = E 0 , then h = 0 and there is nothing to prove. If E 1 = 0, then the metric g 1 satisfies ∇ 2 R = 0 and the orthogonal part of its holonomy algebra h 1 ⊂ so(E 1 ) is irreducible. We will show that this is not possible.
Thus, suppose that (M, g) satisfies ∇ 2 R = 0 and its holonomy algebra equals to g = h + p ∧ E, where h ⊂ so(E) is irreducible.
Then the subspace ∇R(g) 0 ⊂ ∇R(g) of g-annihilated tensors is the one-dimensional subspace given by Proof. Let S ∈ ∇R(g) 0 . For any v ∈ V , the element S v ∈ R(g) can be written as S v = R (R 0 v ,Pv,Tv ) as it is explained above. Since S · (p, ·) = 0, by the second Bianchi identity S p = 0. The fact that g annihilates S can be expressed as Hence, R 0 U (Y, Z)X = 0, i.e. R 0 U = 0. Next, i.e. 2P Z (Y )X = P Z (X)Y . Since this equality holds for any X, Y ∈ E, we conclude P Z = 0. We have got From the second Bianchi identity We conclude P q (Y )X − P q (X)Y = 0. This and the definition of the space P(h) imply P q = 0. Consequently, This implies AT q (X) = T q (AX), i.e. T q commutes with h. Since T q is a symmetric endomorphism of E and h ⊂ so(E) is irreducible, by the Schur Lemma, T q is proportional to the identity. This proves the lemma. We write the metric g in the form (3.1). Then ∂ v is parallel and ∂ v H = 0. By Lemma 3, ∇R has the form for some smooth function f . It is clear that Proof. It is known that Id is the Schouten tensor, Ric is the Ricci operator, and s is the scalar curvature. Recall that by definition, For any vector field U it holds Let the indexes a, b run from 0 to n + 1, and let X 0 = p, X n+1 = q. The covariant derivative of the Ricci operator is given by = f g(p, U 1 )((U 2 ∧ p)p − g ab g(g(U 2 , p)X a − g(X a , p)U 2 , X b )p) = (2 − d)f g(p, U 1 )g(p, U 2 )p.
For the new functionH written as in Theorem 4 we get H kl = H ij a i k a j l , (5.2) 2) implies the existence of an orthogonal matrix a j i such thatH kl is a diagonal matrix with the diagonal elements λ 1 , ..., λ n such that λ 1 ≤ · · · ≤ λ n .
Since ∇R = 0, Corollary 3 shows that H ij is not zero. The transformation (5.1) does not change the form of the metric from Theorem 1 if and only if H kl a k i a l j = H ij and b i (ũ), d(ũ) satisfy certain conditions. This and (5.3) prove the last claim of the theorem.