lorentzian manifolds ;igd;tting a killing vector

Pergamcm Nonlinear Analysis, Theory, Methods & Applications, Vol. 30, No. 1, pp. 643-654, 1997

Proc. 2nd World Congress of Nonlinear Analysts Q 1997 Ekvier Science Ltd

PII: SO362-546X(97)00041-2

Printed in Great Britain. All tights reserved 0362-546X/97 $17.00+0.00

LORENTZIAN MANIFOLDS ;IgD;TTING A KILLING VECTOR

MIGUEL SANCHEZ Departamento de Gcometria y Topologia, Facultad de Ciencias, Univcrsidad de Granada,

lSO71-Granada, Spain

Key words and phrases: Lorentzian manifolds, Killing vector fields, gcodcsic compietencss, geodesic con- nectedacss, cl+scd geodesics, periodic trajectories, isometry groups, Lorentzian torus, Bocimer’s technique, Einstein Lcrentzian manifolds.

1. INTRODUCTION AND GENERALITIES

Lorentzian manifolds admitting a Killing vector field have been studied in the literature from different points of view. Functional analysis, proper actions of Lie groups or Bochner’s technique in Differential Geometry, have been some of the very different tools involved to study them. The pur- pose of this paper is to review some of their properties, pointing out several techniques and supplying references. We will discuss in the next sections several properties of geodesics, isometry groups and Lorentzian tori, and some classification results. Lorentzian manifolds with few (or none) assumptions on curvature will be considered. So, we will not go through the more specific techniques developed to study space forms. (This topic has been widely studied, especially in dimension 3, and we will make just some comments about it in the last section.) In the remainder of Section 1, some notation and general properties of Killing vector fields on Lorentzian manifolds are introduced. The author acknowledges to Prof. P.E. Ehrlich for bringing to his attention some references. This work has been partially supported by a DGICYT Grant No. PB94-0796.

Let (M,g) be a n-dimensional Lorentzian manifold, that is, a (smooth, connected) manifold M

endowed with a non-degenerate metric g with index one (-, +, . . . , +). The causal character of a tangent vector Xp E TpM, p E M will be called timelike (resp. null, spacelike) if g(Xp,Xp) < 0

(rev. dXp,Xp) = 0, 9(xplxp) > 0). U sual concepts in Causality Theory, as time-orientation, future and past, or Lorentzian distance are defined as standard, [3], [28], [40]; we will call spacetime

to a Lorentzian manifold endowed with a time-orientation. A vector field K on M is Killing if it satisfies any of the next well-known alternative conditions:

(i,, its local fluxes consist of isometries, (ii) the Lie derivative of g in its direction is 0, or (iii) g(VXK, Y) = --g(VyK, X), where V denotes the Levi-Civita connection of g, and X, Y are vector

fields. Given the Killing vector field K and a geodesic -y, it is well-known that the product g(r’, K)

is constant (simply, note that its derivative along -y is g(V,,K,T’), which vanishes by the skew- symmetric property (iii)).

When Lorentzian manifolds are studied, the causal characters of its Killing vector fields become relevant. It is very easy to construct examples in which a Killing vector field K reaches exactly one, two or the three causal characters on M (see the metric (4.1) below). Nevertheless, there are some

restrictions when K has zeroes, as the next result shows [4, Lemma 3.21:

PROPOSITION 111. Let (M, g) be a Lorentzian manifold with a non-spacelike (at any point) Killing vector field K. I f Kp = 0 for some p E M then K G 0 in all of M.

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Idea of the proof. For each q E M consider a timelike broken geodesic 7 joiningp and q. To check that K vanishes at q note that: (a) as 7’ is timelike, all its orthogonal vectors are spacelike, but the zero, and (b) as g(r’, K) is constant in each differentiable piece, it must be null. Cl

Moreover, the next result can be deduced first in Lorentz-Minkowski spacetime lL” and then, by lifting the Killing vector field to the Lie algebra of the Lorentz group 01(n), in general [4, Proposition 3.51:

THEOREM 1.2. Let K be a Killing vector field on (M, g) admitting an isolated zero at some p E M. Then the dimension n is even and K becomes timelike, spacelike and null at each neighborhood of p.

The timelike character is the most important causality condition for K from both, the mathematical and the physical point of view. Mathematically, the existence of a timelike Killing vector field K (on the manifold or on its timeorientable double covering) is a useful tool to study geodesics and other topics, providing interesting information on the structure of the manifold, as we will see. From a physical point of view, timelike Killing vector fields model observers travelling by their integral curves which see a non-changing metric. They include well-known relativistic spacetimes as the outer Schwarzschild, Reissner-Nordstriim and Kerr ones.

In fact, we will take several names from General Relativity. Thus (M, g) will be called stationary if it admits a timelike Killing vector field K, and standard stationary if M is a product manifold M = R x S (113 the set of real numbers, S any manifold) and g can be written as:

where r~ and rrs are the natural projections on W and S, resp., dt2 denotes the usual metric on I& and gs, p, w are, resp., a Riemannian metric, a positive function and a l-form, all on S (the time-orientation given by at is chosen canonically, when necessary). As particular cases, when K is irrotational (i.e. its orthogonal distribution is involutive) the stationary manifold is called static, and when w vanishes on S, the standard stationary manifold is called standard static. Locally, every stationary or static Lorentzian manifold looks like the corresponding standard one, with K = 6’t (see, for example, [46, 7.21).

We also consider the case when the Killing vector field is not timelike (at one point), which is expected to provide interesting information about the Lorentzian manifold mainly in low dimensions (see Section 4 and the Remark (3) to Theorem 5.2).

2. GEODESICS

We are meaning by a geodesic on (M, g) a ( necessarily smooth) curve y with parallel velocity, V,,y’ = 0. As a classical characterization, a curve y : [0, l] --+ M joining two given points p, q E M is a geodesic if and only if it is an extremal point of the energy functional:

gb’(s)r p’(s)& (2-l)

for any curve p : [0, l] -----f M joining p and q. The next two difficulties arise for the Lorentzian case, in opposition to the Riemannian one: (A)

even when M is compact, the velocity of a geodesic may not remain in a compact subset of the tangent bundle TM, which is a difficulty to obtain geodesic completeness, and (B) the functional I is not bounded from below and does not satisfy Palais-Smale condition, which yields problems for geodesic connectedness and the existence of closed geodesics or t-periodic trajectories. Frequently, the single fact that g(r’, K) is constant, will be the key to skip the difficulties of the geodesics in the Lorentzian case.

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2.1. GEODESIC COMPLETENESS. Note first that, for Lorentzian manifolds, there is no any analogous conclusion to those of classical Hopf-Binow theorem, and even compact Lorentzian manifolds may be geodesically incomplete (Clifton-Pohl torus [40, Example 7.161 is a well-known counterexample). In our ambient, the next result holds [44]:

THEOREM 2.1. A compact stationary manifold (M, g) is geodesically complete.

Idea of the proof. Use that g(K, 7’) is constant to show that 7’ remains in a compact subset of TM. This implies that the integral curves of the geodesic vector field on TM are complete. Cl

Remarks. (1) If K were just non-spacelike, the result may not hold. On the other hand, in Section 3 we will discuss the differentiable structure of M under the hypothesis of the Theorem 2.1. All these results can be extended to compact manifolds admitting a timelike conformal vector field [44] and, with some additional hypothesis [42], to non-compact manifolds. In no one of these extensions 7’ must remain in a compact subset of TM; nevertheless, it is checked in their proofs that the restriction of 7’ to any bounded interval does remain, which is enough.

(2) A Lorentzian space form is usually defined as a complete Lorentzian manifold of constant curvature. So, in the compact case, the first natural question which appears is if the assumption on completeness is necessary. Some techniques involving the developing map of an affie manifold, commonly used in the study of space forms, have yielded, among others: (A) a remarkable result without any assumption on Killing vector fields: compact flat Lorentzian manifolds are complete [13], and (B) an alternative proof of the Theorem above, with the additional hypothesis for g of having constant curvature, [30]. For more results on geodesic completeness, see the review [47].

2.2. GEODESIC CONNECTEDNESS. As general facts on geodesic connectedness in Lorentzian manifolds we have:

(i) Even a complete or compact Lorentzian manifold may be non-geodesically connected: the de Sitter spacetime (pseudosphere ST) and the Clifton-Pohl torus are counterexamples of each case [40, p. 150, 2601. As far as we know, it is an open question if both conditions together (compact and complete) imply geodesic connectedness.

(ii) Given two points of a Lorentzian manifold, it is relevant to know not only the existence and multiplicity of geodesics joining them but also their causal character. This problem becomes especially interesting because of the different physical interpretation of timelike, null, and spacelike geodesics.

(iii) Consider the important relativistic family of globally hyperbolic spacetimes, that is, time- oriented Lorentzian manifolds admitting a Cauchy hypersurface (topological hypersurface which is crossed exactly once by any inextendible timelike curve, see [3] or [40] for details). By a classical Avez-Seifert result [3, Theorem 2.141, if a point q of such a spacetime lies in the causal future of another one p (that is, they are equal or can be joined by a curve which is non-spacelike and future directed at all its points) then they can be joined by a non-spacelike geodesic, which, moreover, can be chosen with a length non smaller than any non-spacelike future-directed curve joining them.

In our ambient, the next result holds [37, Theorem 3.4.31 (see (81, [9] for previous results):

THEOREM 2.2. Let (M, g) be a standard stationary spacetime, as in (l.l), such that: (a) the metric gs is complete, (b) 0 < InfP, Sup/3 < oo, and (c) w (or th e vector field metrically associated to it) has bounded norm. Then (M, g) is geodesicaily connected.

Idea of the proof. Consider a geodesic 7 : [0, l] -+ W x S, 7(s) = (t(s), I(S)), Vs E [O,l], joining two points (tl, $I), (t2,z2) E M. As the product g(7’, &) is a constant X, a first integral for the component t(s) is obtained. Moreover, the value of X can be computed from t(1) - t(0) = t2 - tl. Thus, the values of t(s) and X can be substituted in the energy functional I, yielding so a new


functional J for the component z(s). This component can be seen as an extremal curve of J, which is a non-local and complicated functional, but bounded from below and satisfying Palais-Smale condition. 0

Remarks. (1) Under the stated hypothesis, (M,g) becomes globally hyperbolic. The variational approach needed to prove Theorem 2.2 allows one to deal with spacelike geodesics, which can not be studied with Avez-Seifert’s techniques. But, if we restrict our attention to manifolds under the hypothesis of Theorem 2.2, Avez-Seifert result can neither be deduced as a corollary of this Theorem: note that, given two causally related points, the geodesic joining them yielded by Theorem 2.2 may be spacelike. Nevertheless, the variational approach can also be used to obtain information on timelike geodesics, see [51] for a discussion.

(2) Some multiplicity results and extensions of Theorem 2.2 (to splitting type manifolds or manifolds with boundary) have been obtained, see [37] for a systematic development.

2.3. CLOSED GEODESICS. Among the most classical problems in Riemannian geometry is the one of establishing the existence of closed geodesics in a Riemannian manifold. Nevertheless, there are few results on this topic in Lorentzian geometry; in fact, the basic question: must a compact Lorentzian manifold have a closed geodesic?, remains open, as far as we know.

For spacelike closed geodesics, the next result holds [36]:

THEOREM 2.3. A standard stationary manifold (M = R x S,g) with compact S admits a closed spacelike geodesic.

Remarks. This result can be seen as a generalization of the classical one: a compact Riemannian manifold admits a closed geodesic. In its proof, the Killing vector field & is used again to change the initial energy functional by a new functional for the projection of the curves on S, which is bounded from below and does satisfy Palais-Smale condition. Some conclusions on the number of closed geodesics are also obtained.

For non-spacelike closed geodesics some remarkable results have been obtained in [21], [22], [53]. In these references, the maximizing properties of the non-spacelike geodesics for the Lorentzian distance in spacetimes are exploited. The basic assumption is to consider Lorentzian manifolds which are compact; no assumption on Killing vector fields is done. It is worth mentioning about them: (A) in [22] Lorentzian tori are shown to have a timelike or null closed geodesic, and a counterexample is given to show that, in dimensions greater than 2, a compact Lorentzian manifold may have no any of such closed geodesics (nevertheless, this counterexample does have closed spacelike geodesics), (B) in [21] free t-homotopy classes (classes of timelike curves which are freely homotopic by means of timelike curves) are studied; under the natural assumption of having a stable t-homotopy class H, a closed timelike geodesic is shown to appear in H, and (C) in [53], the next result, which can also be reobtained by using the techniques in [21], is proven: a compact Lorentzdan manifold admitting a regular Lorentzian covering with a compact Cauchy hypersurface has a closed timelike geodesic.

2.4. T-PERIODIC TRAJECTORIES. From the point of view of the causality, the existence of timelike closed geodesics (or curves) is not desirable. In some Lorentzian manifolds with a good causal behavior the t-periodic trajectories play a role with certain analogies to the closed geodesics.

DEFINITION 2.4. A geodesic in a stationary standard manifold 7 : W + M = W x $7(s) = (t(s),z(s)),Vs E W is a t-periodic trajectory of universal petiod T E W and proper period b > 0 provided that:

t(b) = t(0) + T, t’(b) = t’(O), z(b) = r(O), z’(b) = z’(0).


When there is nol possibility of confussion, 7 is also called T-periodic trajectory.

Remarks. (1) In the static case, the timelike t-periodic trajectories are the relativistic versions of the closed trajectories followed by a particle under a (gravitational) potential in Classical Lagrangian Mechanics. Each, period of these trajectories splits now in two ones: the proper period, related to the proper time in which the particle gives a round, and the universal period [49].

(2) A t-periobic trajectory 7 can be also regarded as a geodesic which projects onto a closed geodesic in the quotient (R x S)/G, where G is the group of isometries generated by the universal period: (t, Z) -~ (t +T, I), V(t, z) E W x S. Thus, the study of t-periodic trajectories can be regarded as a particular ce of the study of closed geodesics.

These trajectories were first introduced for static spacetimes with S = W3 in [7] (even though the definition above is taken from [49]), and after they have been developed in different situations, among them: (i) static chse, timelike trajectories with compact S are studied in [26] , timelike and spacelike trajectories with’s = llV in [9], and some general properties in [49], (ii) general stationary case, under different conditions on M, timelike trajectories are studied in [35], the null ones in [12] and timelike and null~ones in [50].

In these references, the existence of t-periodic trajectories is obtained under certain reasonable hypothesis, and different techniques and points of view are devoloped. So, we have: (A) variational methods and Ljusternik-Schnirelman theory is used in [9], [12] and [35], (B) a natural correspon- dence between geodesics in static spacetimes and closed trajectories in some Lagrangian systems on Biemannian marjifolds is developed in [49], (C) a variant of a shortening procedure used to study closed geodesics in Biemannian manifolds is carried out in [26], and (0) t-periodic trajectories are studied as closed geodesics (as commented in Remark (2)), and the maximizing properties of the non-spacelike geodesics for the Lorentzian distance function are applied in [50]. For more results on t-periodic trajectories in other ambient spaces, we refer to [2], [27], [38], (391, and references therein.

3. COMPACTNESS OF THE ISOMETRY GROUP

It is well-known that the group of the isometries Iso(M, g) of a Riemannian or Lorentzian manifold (M,g) is a Lie group. Moreover, fixed a point p E M and an orthonormal basis B, c TpM, then Iso(M, g) is naturally identifiable to a closed submanifold C of the orthonormal bundle of (M, g), which is obtained as the orbit of BP by means of the elements of Iso(M, g) (see, for example, [31]).

Now, assume that M is compact. I f g is Riemannian, clearly C is compact, and so is Iso(M, g). But if g is Lorentzian, Iso(M, g) may be non-compact. To check it is surprisingly easy:

COUNTEREXAMPLE 3.1. Consider on W2 the flat Lorentzian metric h = dx~+dxl@dx2+dx2@dxl- dx$, in usual coordinates, which is naturally inducible on the canonical torus T2 = B22/Z2. Then the matrix

( 1

1 2 2 5

yields an isometry for h which is also inducible on T 2. But clearly the closed subgroup {An : n E Z} c Iso(T2, h) is not compact, as required.

The next result for Killing vector fields holds:

THEOREM 3.2. Let (M, g) be a compact Lorentzian manifold admitting a Killing vector field K: (1) If K is timelike in a point p E M and the dimension of Iso(M, g) is one, then Iso(M, g) is

compact. (2) If K is timelike everywhere, then the one-parameter subgroup cp associated to K has compact

closure in Iso(M, g), M admits a circle action without fixed points and, in dimension 3, it is a Seifert manifold.


Idea of the proof. (1) Note first that QtK = fK,V@ E Iso(M,g), because of the dimension 1 and KP is a non-null vector. From this fact and the one that K, is timelike, it is easy to prove that the orbit of any orthonormal basis containing a vector proportional to Kp is compact [48, Lemma

4.41. (2) A direct computation shows that K is also Killing for the Riemannian metric obtained as

SR(K,K) = -dK, K), g&,X) = 0, im(X,X) = s(X,X), f or all X g-orthogonal to K. Then, as the group of isometries Iso(M,gR) is compact, the closure of cp in Iso(M,gR) is compact too. Identifying the isometry groups of g and gR with two closed subsets C, CR of the reference bundle of M, ‘p lies in C (7 CR. So, the closures of (o in Iso(M, g) and Iso(M, gR) are identified with the same subset, which yields the first assertion. For the remainder, note that, as Iso(M,gR) is compact and infinite, it must contain a circle S1 as subgroup (see, for example, [41, p. 2511). 0

Remarks. (1) A remarkable result in [15] h s ows: a compact, simply connected and analytic Lorentzian manifold has a compact isometry group. It is not known if the hypothesis of analyticity (imposed there for technical reasons) is necessary, but the simple connection is it, as the counterexample above shows. Thus, the strong assumption on the fundamental group for this result is changed by a strong assumption on the dimension of the isometry group in the first part of Theorem 3.2.

(2) In the proof of the assertion (2) in Theorem 3.2, the circle S1 can be chosen “close” to cp to yield another timelike Killing vector field. Thus, when the action is free, M is a circle bundle on a Riemannian manifold (N, hR), and the metric g controls its curvature, [44].

Finally, it is worh pointing out that the isometries associated to non-spacelike Killing vector fields can be studied in the more general setup of the non-spacelike future isometries. An isometry 4 of a spacetime is called a non-spaceZike future isometry if, for each p, its image q%(p) belongs to the causal future (see Section 2.2) of p. Some properties of these isometries have been studied in [4], [5], [16]. For example, assume that such a 4 admits a fIxed point p; when the spacetime has a good causal behaviour, the reversed triangle inequality can be suitably applied to a normal convex open neighborhood of p, yielding [4, Lemma 3.11: f or strongly causal spacetimes, a future non-spacelike

isometry has no fied points unless it is the identity. (Recall that a spacetime is said to be strongly

causal if every point has arbitrarily small neighborhoods which no nonspacelike curve intersects more than once.) Note that an infinitesimal version of this result was obtained in Proposition 1.1. As a direct consequence, we have [5, Proposition 3.31: the only future nonspacelike isometry of de Sitter

spacetime is the identity.

4. STRUCTURE OF LORENTZIAN TORI

Up to now, we have frequently imposed to the Killing vector field the timelike causal character, but no general assumption on dimensions has been done. Now, we are going to consider dimension two, and no assumption on the causal character of the Killing vector field will be assumed. A compact surface can support a Lorentzian metric if and only if its Euler’s characteristic is 0. So, in the compact case,

only tori must be considered, up to an orientable covering. We will focus our attention on Lorentzian tori, where the existence of a (non-trivial) Killing vector field has been used to give a classification of them [48] (this is our basic reference in Section 4, see this reference for detailed proofs and related results). We refer to [33] and [52] for general properties of Lorentzian surfaces without assumptions on Killing fields, and to [14], [20], [43] f or other properties of Lorentzian tori.

First, we are going to construct an obvious family of Lorentzian tori admitting a Killing vector field, and to study some of their properties. After, we will see that, in fact, this family includes all Lorentzian tori with a Killing vector field, up to coverings.

Consider, in usual coordinates, the metric on R2

g(zl,z2) = E(xl)dx: + F(xl)[dzl ~3 dx2 + dxg @ dxl] - G(xl)dx;, (4.1)


where E, F, G are functions on W satisfying the conditions: (i) EG + F* > 0, (g is Lorentzian) and (ii) E, F and G are periodic with the same period 1 (the metric is naturally inducible on a torus T* by using unid translations). Note that for these metrics the coordinate vector field K = a,, is Killing, and its cbusal character depends on the sign of G.

PROPOSITION 4.1. The next conditions are equivalent for the constructed tori: (i) g is complete in a causal sense (i.e. for timelike, null or spacelike geodesics), (ii) g is complete in the three causal senses, (iii) the coef&ient G has constant sign (identically positive, null or negative); that is, the coor-

dinate Killing vector field K has a definite causal character, and (iv) g is (globally) conformally flat.

Idea of the proof. Note that if y is a geodesic then g(r’, 7’) and g(y’, K) are constant. This fact allows one to integrate the geodesic equations in dimension 2, and, then, to prove the equivalence among the three $rst conditions by brute force. On the other hand, a conformally flat Lorentzian torus is always geodesically complete [44], which yields (iv) + (ii). For (iii) =+ (iv), note that, when G is null, the metric 4 is flat; otherwise, K is also Killing for the conformal metric g* = (l/ ] g(K, K) l)g, and g* is flat. Cl

Call @-tori to those ones constructed above which are conformally flat but non-flat (that is, G is non-constant and ] G ]> 0), and @-tori to the incomplete ones. The next result, in particular, permit us to reduce the study of Lorentzian tori with a Killing vector field to one of these two families.

THEOREM 4.2. The dimension of the isometry group Iso@‘, g) of a Lorentzian torus (Z’*, g) is less or equal to 2, with equality if and only if g is flat. If (Z’*, g) admits a (non-trivial) Killing vector field K, then K does pot vanish at any point and:

(A) (T*, g) is flat if and only if g(K, K) is constant. (B) (T*, g) is iconformally flat if and only g(K, K) h as a definite sign. Moreover, it is conformally

flat but non-flat if and only if it can be covered by a V-tori. (C) (Z’*,g) is,non-conformally flat if and only it can be covered by a @-tori. Even more, in the non-flat cases, Iso@‘*, g) is compact.

Idea of the proof. From a computation, every (non-trivial) Killing vector field K on a Lorentzian surface has isolated zeros with index -1. Thus, K can not vanish on a torus by Poincar&Hopf theorem, and there are at most two independent Killing vector fields. Moreover, if two such independent Killing vector fields K and K’ exist, their fluxes can be used to carry the curvature of each point p to a neighborhood of ip. Thus, in this case, the torus is of constant curvature and, by the Lorentzian Gauss-Bonnet theorem [l] (see also [lo] or [29]) it is flat. This proves the two first assertions, and the necessary condition in (A) follows easily. For the converse, one can use the dimension 2 to check that K is parallel, and there exist another independent parallel vector field.

The last assertion is a consequence of (A) and the Theorem 3.2(l). So, the integral curves of K yield a foliation by circles of T * . This fact and the Proposition 4.1 yields (B) and (C). q

Remarks. In the case (B), the Lorentzian torus is geodesically connected (apply Avez-Seifert result to its universal Lorantzian covering). It is not difficult to find geodesically connected and disconnected examples in the case (C) (see [48, Section 51). In fact, Clifton-Pohl torus (which admits a Killing vector field and is incomplete, lieing in (C)) is an example of non-geodesically connected torus, as commented in Section 2.2.

On the other ~ hand, the Theorem 4.2 and the manageable expression of the metrics (4.1), yield many properties on Lorentzian tori admitting a Killing vector field (see [48, Structure Result], and related consequences).

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5. SOME CLASSIFICATION RESULTS: BOCHNER’S TECHNIQUE

5.1. CLASSIFICATION RESULTS. In Section 4 we characterized Lorentzian tori admitting a Killing vector field without any additional assumption. To obtain classification results in higher dimensions, it seems inavoidable to impose some additional conditions on, for example, curvature. The more extreme assumption occurs in space forms. There are many results on this topic, and some specific techniques have been developed to deal with them. So, compact three dimensional space forms are identified in [19], [23], [24], [32], [34], and their Killing vector fields are widely studied in [56]; we also refer to [IS], [25], [30] and references therein for more information on space forms of higher dimensions. We will focus our attention on manifolds which are stationary; this assumption is enough strong to deal with a less restrictive hypothesis on curvature, considering Einstein manifolds. These manifolds have been studied in [45] by using a variation of classical Bochner’s technique for Riemannian manifolds. In Section 5.2, we will comment the so obtained main results, and compare them with those obtained by using more specific techniques for space forms. In Section 5.3, the difficulties an additional tricks to applicate Bochner’s technique in the Lorentzian case are explained. We refer to [45] for the proofs and more accurate results (including non-timelike Killing vector fields). For the sake of completeness, in the remainder of Section 5.1 we will comment very briefly a remarkable splitting result for manifolds with non-negative timelike curvatures [5, Theorem 5.21:

THEOREM 5.1. Let (M, g) by a complete, simply connected and stationary manifold with nonnegative sectional curvatures for timelike planes. Then (M, g) splits off as a product M = l&H with metric g = -dt* @ AR, where (H, hi) is a complete Riemannian manifold.

Remarks. (1) For any non-spacelike isometry 4 (see the end of Section 3), one can define the displacement function p ---+ d(p, e(p)), where d is the Lorentzian distance. The properties of this function has been studied in [4], [5], [16], and it is possible to obtain the next infinitesimal result [5, Proposition 5.1): any timelike Killing vector field K on a timelike or null complete Lorentzian

manifold of nonnegative timelike curvatures is parallel (VK E O).) Thus, under the hypothesis of the Theorem, K is parallel and the result follows from Wu’s Lorentzian version of de Rham’s Decomposition Theorem [54, p. 2961.

(2) The analogous displacement functions on Hadamard manifolds (complete simply connected Riemannian manifolds of nonpositive sectional curvature) plays an important role in the study of these manifolds and their quotients by discrete groups of isometries. (Recall that, with our sign conventions, negative curvature in the Riemannian case corresponds to positive sectional curvature for timelike planes in the Lorentzian one.) In fact, it follows from results in [ll]: a Killing vector-field

on a complete Riemannian manifold of nonpositive sectional curvature and finite volume is parallel. Note that, for the corresponding Lorentzian results, no assumption on volume is imposed.

(3) The Theorem 5.1 may be compared with the next one for nonpositive curvature, obtained by quite different techniques [6]: Let (M,g) b e a globally hyperbolic spacetime with everywhere non-

positive timelike sectional curvatures. Assume either (a) (M,g) d t a mi s a complete timelike geodesic

line, or (b) (M, g) admits a compact Cauchy surface and is timelike complete. Then (M, g) splits

ofl as in the Theorem 5.1. (Recall that a timelike geodesic is a line if it maximizes absolutely the Lorentzian distance.)

5.2. EINSTEIN MANIFOLDS. The main result obtained for timelike Killing vector fields in [45] is:

THEOREM 5.2. Let (M,g) be a n-dimensional compact stationary Einstein manifold, and K a

timelike Killing vector field. (A) The scalar curvature of g is non-positive. (B) If g is Ricci-flat then K is parallel, the first Betti number of M is not zero and the Levi-


Civita connection of g is Riemannian (i.e. there exist a Riemannian metric with the same Levi-Civita connection of g).

(C) If g is Ricci-flat and also satisfies one of the three following conditions: (i) ,(M, g) is homogeneous, (ii) (M, g) is flat,

then (M,g) is isometric to a flat n-torus, up to a covering. or (iii) n 5 4,

Remarks. (1) One of the three conditions in (C) is necessary, because there exists counterexamples for each n 1 5. There are also Lorentzian examples of: (i) Einstein compact manifolds with positive scalar curvature, (ii) Ricci-flat non-flat homogeneous compact manifolds, and (iii) flat compact n- manifolds (n 2 3)~ which can not be covered by a flat Lorentzian n-torus. On the other hand, any Killing vector field on a compact, flat, homogeneous Lorentzian manifold must be parallel [17], and, thus, these manifoilds must be n-tori.

(2) By quoted de Rham-Wu’s result [54], the conclusion (B) implies that (M, g) can be covered by a standard sta t ic manifold W x S where (S, gs) is a Ricci-flat Riemannian manifold.

(3) Non-timeli/ce vector fields also provides interesting information in dimension 3 (note that in this dimension Einstein manifolds have constant curvature), in particular (see [30, Theorem E(e)], [45, Propositions 3.5, 3.71): Let (M,g) b e a compact S-manifold of constant (sectional) curvature c admitting a (non-trivial) Killing vector field K. If K is spacelike then c 5 0, and if K is null then c = 0 and K is parallel.

By using more specific techniques for space forms the obtained results for the stationary ones, as collected in [30] are:

RESULTS. Let (M[, g) be a compact stationary space form of curvature c. Then: (a) c 5 0, (b) if c = 0 then the first Betti number of M is not zero and the Levi-Civita connection of g is Riemannian, and (c) if c < 0 then M is a circle bundle on a negatively curved compact manifold, up to a finite covering.

Remarks. (1) The results (a) and (b) are very widely improved by using Bochner’s technique. In fact, the Theorem, 5.2 yields a surprisingly easy classijication of flat compact stationary Lorentzian space forms: all oj them are, up to a covering, Lorentzian tori.

(2) As no compact stationary space form of positive constant curvature exist, the problem of the classification is then reduced to the ones of negative curvature, for which the result (c) applies.

(3) As we saw in Section 3, without any assumption on curvature, if the stationary manifold (M, g) is compact, then it admits a circle action without fixed points, and, when this action is free, then M is a circle bundle on a Riemannian manifold (N, hi). By O’Neill’s formula for submersions (see, for example, [40, Theorem 7.471) if (M, g) h as negative curvature, then (N, hR) supports a metric of negative curvature. This extends (c) to more general manifolds.

5.3. BOCHNER'S ?;ECHWIQUE. Bochner’s technique has been widely used in Riemannian manifolds, where the Laplacian is an elliptic operator (take, for instance, 1551 as reference). Now, we are going to explain how Bochner’s technique can be applied in Lorentzian manifolds, which is the main tool to prove Theorem 5.2.

(A) Classical Bockner’s formula. For compact M and any vector field Z on (M,g),

J M(Ricci(Z, 2) + trace(Ai) - (traceAz)*)dv = 0 (5.1)

where A is the endomorphism field 2, ---+ V,Z,Vv E TM. Observe that this formula is valid as if g is a Riemannian or Lorentzian metric. In fact, it just expresses that the integral of the divergence of the vector field X F AZ(Z) - (divZ)Z vanishes.


(B) Dificulty of the Lorentzian case. Put

62 = trace(Ai) - (traceAZ)*.

I f 62 is signed, Bochner’s formula (5.1) can be claimed, yielding a strong link between Z and the sign of the Ricci curvature. In fact, as a classical application in the Riemannian case, assume that 2 = K is a Killing vector field. Then AK is skew-adjoint and 6K 5 0 with equality if and only if K is parallel. Thus, one has: any Killing vector field on a compact Riemannian manifold with Ricci 5 0 is parallel, and if, moreover, Ricci < 0 in a point, it is identically 0.

Nevertheless, if g is Lorentzian and K is a timelike Killing vector field, then the corresponding 6~ is not necessarily signed, even though AK is skew-adjoint. Thus, the technique can not be directly USd.

(C) Trick in the Lore&&an case. This problem in the Lorentzian case can be solved by taking, for each timelike Killing vector field K, the corresponding unitary vector 2 = (-l/g(K, K))l/*K. As 2 is unitary, then AZ can be restricted to the orthogonal of 2, and the corresponding restriction AL of AZ also satisfies:

62 = trace(A2) - (traceAL)*.

Moreover, note that AtZ is again skew-adjoint, and it operates on a positive definite subspace. Then one has 62 < 0, with equality if and only if AL e 0.

Thus, Bochner technique can be applied to 2, even in the Lorentzian case. Then, it is straight- forward to check (A) in Theorem 5.1. Even more, if AfZ is shown to be identically null, one obtains, for the timelike Killing vector field K, that 6~ 2 0. so, Bochner’s formula (5.1) yields information also when applied to K, which can be used to prove (B) and (C).

Remark. In order to apply Bochner’s formula we needed just to have a timelike rigid vector field, that is, a timelike unit vector field 2 such that its corresponding AL is skew-adjoint (46, p. 561. We have used that if K is timelike Killing, then the corresponding unit vector field Z is timelike rigid. But it is not difficult to find compact Lorentzian manifolds with timelike rigid vector fields which does not admit any timelike Killing vector field. In fact, such examples can be found in the incomplete tori constructed in Section 4, [45, Section 21, [48, Remark 6.2).

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