Symmetries of Lorentzian Three-Manifolds with Recurrent Curvature

Locally homogeneous Lorentzian three-manifolds with recurrect curvature are special examples of Walker manifolds, that is, they admit a parallel null vector field. We obtain a full classification of the symmetries of these spaces, with particular regard to symmetries related to their curvature: Ricci and matter collineations, curvature and Weyl collineations. Several results are given for the broader class of three-dimensional Walker manifolds.


Introduction
A Walker manifold is a pseudo-Riemannian manifold (M, g) admitting a degenerate parallel distribution. Such a phenomenon is peculiar to the case of indefinite metrics. As such, it is responsible for many special geometric properties of pseudo-Riemannian manifolds which do not have any Riemannian counterpart, and has been investigated by several authors under different points of view. The monograph [5] is a well-written recent survey on Walker manifolds and the various related research areas.
Lorentzian three-manifolds admitting a parallel degenerate line field have been studied in [12]. These Lorentzian metrics are described in terms of a suitable system of local coordinates (t, x, y) and form a large class, depending on an arbitrary function f (t, x, y). The case of strictly Walker manifolds, where the parallel degenerate line field is spanned by a parallel null vector field, is characterized by condition f = f (x, y). The results of [12] have been recently used in [14] to obtain a complete classification of the models of locally homogeneous Lorentzian three-manifolds with recurrent curvature.
The aim of this paper is to investigate symmetries of these Lorentzian spaces. If (M, g) denotes a Lorentzian manifold and T a tensor on (M, g), codifying some either mathematical or physical quantity, a symmetry of T is a one-parameter group of diffeomorphisms of (M, g), leaving T invariant. As such, it corresponds to a vector field X satisfying L X T = 0, where L denotes the Lie derivative. Isometries are a well known example of symmetries, for which T = g is the metric tensor. The corresponding vector field X is then a Killing vector field. Homotheties and conformal motions on (M, g) are again examples of symmetries. In recent years, symmetries related to the curvature of the manifold have been investigated. Among them: curvature collineations (where T =R is the curvature tensor), Weyl collineations (T =W being the Weyl conformal curvature tensor) and Ricci collineations, for which T = is the Ricci tensor. We may refer to the monograph [16] for further information and references on symmetries. Ricci and curvature collineations have been investigated in several classes of Lorentzian manifolds (see, for example, [1,6,7,8,9,13,15,17,18,19,20,21,22] and references therein). Because of their physical relevance, in most cases curvature symmetries have been studied for some spacetimes. Moreover, the three-dimensional case has also been considered as an interesting source of examples and different behaviours (see, for example, [8]).
A matter collineation of a Lorentzian manifold (M, g) is a vector field X, corresponding to a symmetry of the energy-momentum tensor T = − 1 2 τ g, where τ denotes the scalar curvature. Matter collineations are more relevant from a physical point of view [10,11], while Ricci collineations have a more clear geometrical significance, since is naturally deduced from the connection of the metric [19]. These physical and geometrical meanings do coincide in a special case, namely, for metrics with vanishing scalar curvature. And this is exactly the case for any strictly Walker three-manifold [12].
We shall obtain complete classifications of curvature and Ricci (≡ matter) collineations of homogeneous Lorentzian three-manifolds with recurrent curvature. In Section 2 we shall give some basic information about Walker three-manifolds and curvature symmetries. In Section 3 we then investigate symmetries of an arbitrary strictly Walker three-manifold. Since the function f = f (x, y) determining the metric tensor here is arbitrary, one cannot expect to obtain these symmetries explicitly in the general case. However, we describe the sets of partial differential equations describing the different symmetries and use them to give some explicit examples of proper symmetries. Then, in Section 4 we shall completely classify the symmetries of homogeneous Lorentzian three-manifolds with recurrent curvature. All calculations have also been checked using Maple16 c .

Three-dimensional Walker metrics
We shall essentially follow the notations used in [12]. A three-dimensional Lorentzian manifold M admitting a parallel degenerate line field has local coordinates (t, x, y), such that with respect to the local frame field {∂ t , ∂ x , ∂ y } the Lorentzian metric is given by for some function f (t, x, y). In the above expression, ε = ±1. However, it is easily seen that by reversing the metric and changing the sign of the coordinate x, without loss of generality one can reduce to the case ε = 1 (as it was done, for example, in [14]).
The parallel degenerate line field is spanned by ∂ t , and the existence of a parallel null vector U = ∂ t (strictly Walker metric) is characterized by the independence of the function f of the variable t [23]. Therefore, with respect to local coordinates (t, x, y), the general form of a strictly Walker metric is given by for an arbitrary smooth function f . With respect to the coordinate basis {∂ t , ∂ x , ∂ y }, the Levi-Civita connection ∇ and curvature R of the metric g f described by (2.1) are completely determined by the following possibly non-vanishing components (see also [12]): Y ] . From (2.2) and (2.3), a straightforward calculation yields that the covariant derivative of the curvature tensor is completely determined by the possibly non-vanishing components Either by (2.4) or by direct calculation differentiating the Ricci identity, it is easily seen that three-dimensional (strictly) Walker metrics have recurrent curvature, that is, in a neighborhood of any point of non-vanishing curvature, one has ∇R = ω ⊗ R, for a suitable one-form ω. Since we are interested in the study of the nonflat examples with recurrent curvature, throughout the paper we shall assume that f xx = 0 at any point. In local coordinates (t, x, y), the Ricci tensor of any metric (2.1) is given by A pseudo-Riemannian manifold (M, g) is said to be locally homogeneous if for any pair of points p, q ∈ M there exist a neighbourhood U of p, a neighbourhood V of q and an isometry φ : U → V . Hence, locally homogeneous manifolds "look the same" around each point. For any given class of pseudo-Riemannian manifolds, it is a natural problem to determine its locally homogeneous examples.
Locally homogeneous examples among three-dimensional Walker metrics have been investigated in [14] (see also [3]). Rewriting the classification obtained in [14] in terms of coordinates (t, x, y) used in (2.1), we have the following. 14]). Locally homogeneous Lorentzian three-manifolds of recurrent curvature naturally divide into three classes. They correspond to one of the following types of (strictly) Walker metrics, as described in (2.1):

Curvature and Ricci collineations
Let (M, g) denote a pseudo-Riemannian manifold (in particular, a Lorentzian one). A vector field X on M preserving its metric tensor g, the corresponding Levi-Civita connection ∇, its curvature tensor R or its Ricci tensor , is respectively known as a Killing vector field, an affine vector field, a curvature collineation or a Ricci collineation.
It is obvious that if X preserves g (respectively, ∇, R), then it also preserves ∇ (respectively, R, ), but the converse does not hold in general. Homothetic vector fields (i.e., vector fields X satisfying L X g = λg for some real constant λ) are again necessarily curvature collineations (in particular, Ricci collineations). For this reason, we are specifically interested in the existence of proper Ricci and curvature collineations, namely, the ones which are not homothetic (and hence, not Killing). Thus, we also need to specify which are the Killing, affine and homothetic vector fields, which is an interesting problem on its own, due to the natural geometric meaning of such symmetries.
Conditions defining Ricci and curvature collineations are formally similar to the ones defining Killing or affine vector fields. However, they may show some deeply different behaviours. In fact (see, for example, [16,19]): (a) Killing and affine vector fields are smooth (provided they are at least C 1 ). However, for any positive integer k, there exist Lorentzian metrics admitting Ricci (and curvature) collineations, which are C k but not C k+1 .
(b) Unlike Killing and affine vector fields, Ricci (and curvature) collineations form a vector space which may be infinite-dimensional and (because of the above point (a)) is not necessarily a Lie algebra. In fact, if X, Y are Ricci (curvature) collineations, then [X, Y ] might not be differentiable.
(c) While Killing and affine vector fields agreeing in the neighbourhood of a point must coincide everywhere, two Ricci (respectively, curvature) collineations that agree on an non-empty subset of M may not agree on M , since they are not uniquely determined by the value of X and its covariant derivatives of any order at a point.
Observe that the above item (b), as concerns the possibility of the vector space of Ricci collineations to be infinite-dimensional, refers to cases where the Ricci tensor is necessarily degenerate (as it is always the case, for example, for three-dimensional strictly Walker metrics). On the other hand, if (respectively, T = − 1 2 τ g) is nondegenerate, then Ricci (respectively, matter) collineations form a finite-dimensional Lie algebra of smooth vectors. In fact, in such a case, they are exactly the Killing vector fields of the nondegenerate metric tensor .

vector field if and only if
where f 1 , f 2 are smooth functions on M , satisfying ii) a homothetic, non-Killing vector field if and only if

3)
where η = 0 is a real constant and iii) an affine Killing vector field if and only if Proof . We start from an arbitrary smooth vector field X = X 1 ∂ t + X 2 ∂ x + X 3 ∂ y on the three-dimensional strict Walker manifold (M, g f ), where g f is described by equation (2.1), and calculate L X g f . Then, X satisfies L X g f = ηg f for some real constant η if and only if the following system of partial differential equations is satisfied: We then proceed to integrate (3.7). From the first three equations in (3.7) we get X 2 = η 2 x − a 1 (y)t + f 1 (y) and X 3 = a 1 (y)x + b 1 (y). Then, the fourth equation in (3.7) yields X 1 = ηt − a 1 (y)tx − b 1 (y)t + f 4 (x, y). Substituting this into the fifth equation, we get which must hold for all values of t, implying that a 1 (y) = c 1 is a constant. Now, the last equation in (3.7) gives which immediately yields that c 1 ∂ x f + 2b 1 (y) = 0 and so, c 1 ∂ 2 xx f = 0. Since we assumed ∂ 2 xx f = 0, we then have c 1 = 0 and integrating b 1 (y) = 0 we get b 1 (y) = c 2 y + c 3 . On the other hand, from the fifth equation in (3.7) we now have f 4 (x, y) = −f 2 (y)x + f 5 (y) and the last equation gives This proves the statement i) in the case η = 0 and the statement ii) if we assume η = 0. With regard to affine vector fields, expressing condition L X ∇ = 0 in the coordinate basis {∂ t , ∂ x , ∂ y }, we get the following system of partial differential equations: As for the above system (3.7), we then proceed to integrate (3.8). From the first equation we get X 3 = c 1 t + a 1 (y)x + f 2 (y) and then the fifth equation yields 2a 1 (y) = c 1 ∂ x f , so that c 1 ∂ 2 xx f = 0 and so, c 1 = 0. Then, a 1 (y) = c 2 is a constant.
Integrating the third and fourth equations (taking into account the first one) we get X 1 = f 3 (y)t + f 4 (x, y), X 2 = c 3 t + f 5 (y)x + f 6 (y). The sixth equation then gives f 3 (y) + c 3 ∂ x f = 0, which, by the same argument above, yields c 3 = 0 and f 3 (y) = c 4 .
By the ninth and tenth equations we then have f 6 (y) + f 7 (y) + 1 2 (2c 7 − c 4 − c 5 )∂ x f = 0, so that c 7 = c 4 +c 5 2 and f 7 (y) = −f 6 (y) + c 8 . Integrating the tenth equation with respect to the variable x, we get We differentiate the above equation with respect to y and subtract the eleventh equation, obtaining f 9 (y) − 2f 8 (y) = c 8 ∂ x f , which immediately leads to c 8 = 0 and f 9 (y) = 2f 8 (y) + c 9 . The statement follows after we suitably rename the remaining constants and functions.   3 (homothetic fixed points). The existence on a Lorentzian manifold (M, g) of homothetic fixed points, that is, of a non-trivial homothetic vector field X which vanishes at a point m ∈ M , has some important consequences on the structure of the manifold itself. Different conclusions can be deduced depending on whether m is an isolated fixed point or not. In the latter case, the zeroes of X form a null geodesic, and the resulting metric is a kind of plane wave, whose conformal vector fields can be determined. Interesting studies of the link between homothetic and conformal vector fields (and their fixed points) and the geometry the metrics can be found in [2,4,17,18]. The above Theorem 3.1 and the special cases described in Theorem 3.6 and in Section 4, allow us to discuss the existence of homothetic fixed points for all three-dimensional Walker metrics, and gives a unified treatment for a large class of threedimensional manifolds, where all different behaviours can occur, from metrics with no proper homothetic vector fields, to cases where homothetic fixed points occur and can be explicitly determined.
We now turn our attention to curvature collineations and prove the following. Theorem 3.4. Let X = X 1 ∂ t + X 2 ∂ x + X 3 ∂ y be an arbitrary smooth vector field on the strictly Walker manifold (M, g f ), where g f is described as in (2.1). Then: i) X is a Ricci collineation if and only if one of the following cases occurs: (a) f is arbitrary and where f 1 is an arbitrary smooth function on M , and the Ricci collineation is defined in the open subset where ∂ 3 xxx f = 0.
ii) X is a curvature collineation if and only if X is a special Ricci collineation of one of the following types: Proof . Because of equation (2.5), a smooth vector field X = X 1 ∂ t + X 2 ∂ x + X 3 ∂ y on a strictly Walker manifold (M, g f ) is a Ricci collineation if and only if As we already mentioned, we are always assuming that ∂ 2 xx f = 0. Consequently, from the first two equations in (3.9) we have X 3 = X 3 (y), and the third equation becomes In the open subset where ∂ 3 xxx f = 0, from the above equation (3.10) we get at once the case (a). Case (b) is obtained as a special solution of (3.10), assuming that ∂ 3 xxx f = 0. We then consider curvature collineations, starting from an arbitrary Ricci collineation as described in cases (a) and (b) and requiring the additional condition L X R = 0. Calculations are of the same kind for all these cases. For this reason, we report the details only for case (b).
So, consider a Ricci collineation In particular, calculating the condition L X R = 0 on the pairs of coordinate vector fields ∂ t , ∂ x , ∂ y , we find that X is a curvature collineation if and only if the following equations hold: It easily follows from the first of the above equations that ∂ x X 1 and X 2 are functions of the variables (x, y). Since f 1 (y) = 0, differentiating with respect to x the second of the above equations we get ∂ 2 xx X 2 = 0 and so, X 2 = f 4 (y)x + f 5 (y). Now, again the second equation gives X 1 = c 1 f 1 (y) 2f 1 (y) √ |f 1 (y)| t + 2f 4 (y)t + f 6 (x, y). Then, since ∂ x X 1 = −∂ y X 2 , we conclude that x + f 7 (y) and this ends the proof.
Observe that taking X 2 = X 3 = 0, all equations in (3.9) are satisfied. Therefore, X = X 1 ∂ t is a Ricci collineation for any arbitrary smooth function X 1 = X 1 (t, x, y), and (by case (a) ) a curvature collineation for any smooth function X 1 = X 1 (y). This implies at once the following.
Corollary 3.5. For any strictly Walker three-manifold (M, g f ), the Lie algebras of smooth Ricci collineations and smooth curvature collineations are infinite-dimensional. In particular, each of these spaces admits proper Ricci and curvature collineations.
We end this section calculating the symmetries of a locally conformally flat strictly Walker three-manifold. By direct calculations of the Cotton tensor of a strictly Walker three-manifold (M, g f ) (see also [12]), it is easily seen that this manifold is locally conformally flat if and only if ∂ 3 xxx f vanishes identically, that is, when the defining function is of the form f (x, y) = p(y)x 2 +q(y)x+r(y) (with p(y) = 0 in order to avoid the flat case). We now prove the following.
Theorem 3.6. Let X = X 1 ∂ t +X 2 ∂ x +X 3 ∂ y be an arbitrary smooth vector field on a conformally flat strictly Walker manifold (M, g f ), where g f is described as in (2.1) with f (x, y) = p(y)x 2 + q(y)x + r(y) (p(y) = 0). Then, X is: , where f 1 (y) and f 2 (y) are arbitrary smooth functions on M .
This equation immediately proves the second statement, since the coefficients of x and its powers must vanish, in order to satisfy it identically. The first statement now follows by setting η = 0 in the equations of homothetic vector fields. With regard to affine Killing vector fields, X must satisfy equations (3.5) and (3.6). So by straightforward calculations, the functions p(y), q(y) and r(y) must satisfy which leads to the third statement. Assertions (iv) and (v) are direct consequences of the cases (b) and (b) of Theorem 3.4, respectively.
With regard to affine vector fields, setting f (x, y) = −2e bx Next, the result on Ricci collineations follows easily from the fact that they are characterized by equations ∂ t X 3 = ∂ x X 3 = 0, ∂ y X 3 + bX 2 = 0.
In particular, a Ricci collineation is also a curvature collineation when it satisfies f 1 (y) + ∂ t X 1 = 0, 2f 1 (y) − b∂ x X 1 = 0, which proves the last part of the statement.
With regard to homogeneous three-dimensional Lorentzian strictly Walker manifolds of type P c and CW ε , comparing their defining functions f (x, y) with the one of a locally conformally flat strictly Walker three-manifold, it is easy to conclude that these homogeneous spaces are indeed locally conformally flat. Therefore, their symmetries can be deduced as special cases of the results obtained in Theorem 3.6. In this way, we obtain the following.  Let h(y) denote a smooth function explicitly determined from α(y) by equation h (y) + α(y)h(y) = 0.
An arbitrary smooth vector field X = X 1 ∂ t + X 2 ∂ x + X 3 ∂ y on M : • is Killing if and only if