Genericity of Nondegeneracy for Light Rays in Stationary Spacetimes

Digital Object Identifier (DOI) 10.1007/s00220-009-0742-3Commun. Math. Phys. 287, 903–923 (2009) Communications in

MathematicalPhysics

Genericity of Nondegeneracy for Light Raysin Stationary Spacetimes

Roberto Giambò1, Fabio Giannoni1, Paolo Piccione2,�

1 Dipartimento di Matematica e Informatica, Università di Camerino,Via Madonna delle Carceri, 9, 62032 Camerino (MC), Italy.E-mail: [email protected]; [email protected]

2 Departamento de Matemática, Universidade de São Paulo, Rua do Matão 1010,CEP 05508-900, São Paulo, SP, Brazil. E-mail: [email protected]

Received: 10 April 2008 / Accepted: 5 November 2008Published online: 18 February 2009 – © Springer-Verlag 2009

Abstract: Given a Lorentzian manifold (M, g), an event p and an observer U in M ,then p and U are light conjugate if there exists a lightlike geodesic γ : [0, 1] → Mjoining p and U whose endpoints are conjugate along γ . Using functional analyticaltechniques, we prove that if one fixes p and U in a differentiable manifold M , then theset of stationary Lorentzian metrics in M for which p and U are not light conjugateis generic in a strong sense. The result is obtained by reduction to a Finsler geodesicproblem via a second order Fermat principle for light rays, and using a transversalityargument in an infinite dimensional Banach manifold setup.

1. Introduction

Multiplicity of light rays from a light source to a receiver in a Lorentzian manifoldmodels the so-called gravitational lensing effect in General Relativity. This is an activeresearch field in both Physics and Geometry; a growing interest in this research areahas been triggered in recent years by an increasing amount of observational material inAstrophysics. Some living reviews on the mathematical aspects (see [23]) as well as onthe observational aspects (see [25]) of gravitational lensing are available on the web.

Variational techniques apply to the light ray problem, which has a variational naturegiven by the Fermat principle. In particular, Morse theoretical results have been obtainedin several contexts; Morse relations give lower estimates on the number of light raysissuing from a fixed event and terminating on a given observer.

An essential assumption for the Morse theory of a functional f defined on a Hilbertmanifold X is that all its critical points be (strongly) nondegenerate, in which case fis said to be a Morse function. In the light ray case, the functional to be studied is theso called arrival time (or departure time, according to the time orientation). The arrivaltime functional is defined on the set of all lightlike (future or past pointing) curves

� Current address: Department of Mathematics, University of Murcia,Campus de Espinardo, 30100 Espinardo, Murcia, Spain

904 R. Giambò, F. Giannoni, P. Piccione

joining an event p and an observer U in a time oriented Lorentzian manifold (M, g).This is a Morse function precisely when p and U are not conjugate along any lightlikegeodesic. If one fixes the ambient space (M, g), such non-conjugacy assumptions holdgenerically in the set of pairs (p, U ), and this is established easily using the propertiesof the exponential map of (M, g). In this paper we address the question of establishingthe genericity of the non-conjugacy property for a fixed pair (p, U ) when the spacetimemetric varies. This is no longer a finite dimensional problem, and it must be studiedusing an infinite dimensional Banach manifold approach.

Before we get into the details of the result presented, let us discuss the original moti-vations that led us to this research. On one hand, it is clear that in view of physicalapplications, it is always desirable to have results that prove the stability of a theoryby small perturbations of the data. Such data are possibly the outcome of an imperfectobservation procedure in a physical experiment, or the result of some kind of theoreti-cal approximation of the physical model. On the other hand, for the specific genericityproblem discussed in the present paper, the main motivation comes from Morse theory.Unlike its finite dimensional counterpart, infinite dimensional Morse theory uses in anessential way several auxiliary data that do not belong properly to the variational problemitself. For instance, the choice of a specific Hilbert–Riemann structure in (an appropriatemetric space completion of) the trial space, which does not belong specifically to thevariational problem, is essential in order to guarantee:

• the Palais–Smale condition,• the condition of transversality of the stable and unstable manifolds of critical points

(Morse–Smale condition).

As to the transversality of the stable and unstable manifolds, in many examples thisis known to be generic in classes of variational problems having only nondegeneratecritical points. Thus, it becomes an interesting question to establish the genericity ofthe nondegeneracy condition. For the Lorentzian geodesic variational problem, a recentresult due to Abbondandolo and Majer (see [1,2]) shows that, under suitable technicalassumptions, the homology of the Morse complex constructed using the dynamics of thegradient flow, is stable by uniformly (C0) small perturbations of the Lorentzian metrictensor that preserve the non conjugacy property. An analogous result is likely to holdalso in the case of light rays. This implies in particular that the Morse theory of light raysbetween a fixed event p and a fixed observer U is unchanged when the metric tensorvaries in a connected open set containing a C0-dense subset of metrics for which p andU are non conjugate along any lightlike geodesic. With a result of this type at hand,clearly one would reduce the Morse theory for light rays to an analysis of the questionin simple spacetimes, like for instance standard static products, avoiding the technicaldifficulties as encountered in [11–13,21,22] etc.

Generic properties of geodesic flows have been studied mostly in the context of Rie-mannian geometry, see for instance [3,4,15,16]. Recently, B. White [26] has proved agenericity property of the nondegeneracy condition for minimal embeddings of higherdimensional submanifolds. Recently, the genericity of semi-Riemannian metrics forwhich two fixed endpoints are non conjugate has been proven in [8]. Nondegeneracy of(periodic) solutions of general flows on manifolds has been studied by several authors;a classical reference for this topic is the book [5]. As to the class of Hamiltonian sys-tems, in which case the dynamics of the solutions may be different essentially in distinctenergy levels, the nondegeneracy condition may fail to be generic, see [18].

As to the problem studied in this paper, the first observation is that the nonconjugacyproperty is easily stated in terms of transversality (Proposition 2.1, a manifold exten-

Genericity of Lightlike Nondegeneracy 905

sion of the original result of B. White [26, Theorem 1.2]). Nevertheless, for the lightray problem there are two main obstructions to the classical genericity theory. First, theset of trial paths, i.e., the set of lightlike curves from a fixed event to a fixed observer,does not have the appropriate C2-regularity necessary to develop the theory. A second,more subtle, difficulty is the fact that, if one varies the spacetime metric, then it is thedomain of the functional that changes, rather than the functional itself. This suggeststhat in order to study this problem one needs to restrict the class of spacetimes. Moreprecisely, in order to maintain a fixed domain for the arrival time functional when thespacetime metric is changed, a natural assumption is that the spacetime admits a globalsplitting structure. In spacetime of this type there exists a bijection between lightlikecurves and their projection onto the space component, which allows to keep the domainof the arrival time functional independent of the metric tensor. As an initial step for thisresearch, we will consider Lorentzian manifolds (M, g) admitting a global splitting ofthe form M = M0 × R, with spacelike slices M0 × {t}, and with metric invariant bytime translations (p, t) �→ (p, t + t0). These spacetimes form a class called standardstationary. Light rays in stationary spacetimes M0×R are associated naturally to Finslergeodesics in the base manifold M0 (see for instance the recent article [9] with full detailson this topic), where the arrival time functional turns into the Finsler length functional.However, this does not solve the question of regularity of the functional. The Finlsermetric on the base manifold M0 belongs to a special class of metrics called Randersmetrics; Randers metrics are determined by the choice of a Riemannian metric g and a1-form ω of norm less than 1 (see Sect. 3). For this class of metrics, the length func-tional, which is not smooth, can be replaced by a smooth functional Fg,ω (see (3.1)), asit was originally observed in [10]. A crucial point is then to establish that nondegeneratecritical points of Fg,ω correspond to nondegenerate lightlike geodesics in M ; this isproved by means of a second order variational principle (see Sect. 3.3). This principlesays that elements in the kernel of the second variation of Fg,ω at a given critical pointare the spatial component of a Jacobi field along the corresponding lightlike geodesic.The second order variational principle was originally stated and proved by A. Masiello,see [17], using an abstract argument. In this paper we reprove the result by an explicitcomputation of the index form and the differential equation satisfied by elements in itskernel, which is more suited for our purposes.

A general remark on the existence of a second order Fermat principle is in order.The reader should recall that there exist other variational principles relating solutionsof physical systems to geodesics in appropriate manifolds. Two classical examples aregiven by the Maupertuis principle and by the Kaluza–Klein principle. Maupertuis prin-ciple associates solutions of conservative dynamical systems in a fixed energy level togeodesics in conformal metrics. Kaluza–Klein principle associates trajectories in a rela-tivistic electro-magnetic field of charged particles (having a fixed charge-to-mass ratio)to geodesics in a higher dimensional Lorentz manifold which is a principal fiber bundleover the spacetime. The interesting fact is that, in neither of these two cases, a secondorder principle holds, and thus nondegeneracy of solutions cannot be inferred from thenondegeneracy of the corresponding geodesic. In this sense, the Fermat principle forlight rays is an exception.

Using this second order principle, the genericity question is now studied in terms ofthe functionals Fg,ω; these functionals are Fredholm (Proposition 3.2), and the genericityof the nondegeneracy condition for their critical point is established using an abstractcriterion proved in [8,26] (see Proposition 2.1). This criterion involves the second mixedderivative of the functionals, computed explicitly in Sect. 4 (see (4.5), (4.6), (4.7)), and


the existence of certain tensors on the manifold M0 whose value and covariant derivativehave been assigned along a sufficiently small portion of an injective path. The existenceof tensors with these data is established in a technical lemma (Lemma 4.1) proved bytechniques of calculus with connections in affine manifolds.

The main result of the paper, Theorem 4.4, states that, given a manifold M0, the setof standard stationary metrics on M = M0 × R for which a fixed point (p0, 0) and afixed observer U = {p1} × R are joined only by nondegenerate lightlike geodesics isgeneric in a strong sense. More precisely, a standard stationary Lorentzian metric tensoron M0 ×R is determined, up to an irrelevant conformal, by a Riemannian metric tensorg on the base manifold M0 and by the choice of a 1-form ω = g(δ, ·) on M0 (see (4.1));here δ is a smooth vector field on M0. We prove that, for fixed δ, the set of Riemannianmetrics g for which the corresponding Lorentzian metric tensor has only nondegeneratelightlike geodesics between a fixed event (p0, 0) and a fixed observer U = {p1} ×R isgeneric. More surprisingly, we also prove that if one fixes g, then the set of δ’s for whichthe corresponding Lorentzian metric tensor has only nondegenerate lightlike geodesicsbetween a fixed event (p0, 0) and a fixed observer U = {p1} × R is generic.

Let us conclude with some final remarks.First, we point out that we consider the case that the base manifold M0 is not nec-

essarily compact. Note that in the noncompact case, there is no natural Banach spacestructure on the space of tensors of any type, and thus one needs to impose some restric-tions on the growth at infinity of the metrics to be considered. This can be done in several(non canonical) ways. In order to preserve generality of our result, we axiomatize a fewproperties that characterize the type of Banach space of tensors that can be consideredin the genericity result, with the introduction of the notion of an admissible triple oftensors (see Sect. 4.2). This is a very general notion, and it includes all possible typesof Banach spaces of tensors with controlled Ck-growth at infinity.

Second, we remark that the genericity result is only proved in the case that the event(p0, 0) does not belong to the observer U = {p1} × R, i.e., when p0 �= p1, althoughthe result is very likely to be true also in this case. The proof presented in this paperfails in the case p0 = p1 due to a subtle technical point, which arises when one dealswith lightlike geodesics whose spatial component is periodic. Given one such geodesicγ = (x, t) : [0, 1] → M0 ×R with period T = 1/N , N ≥ 2, if there exists a nontrivialJacobi field V = (ξ, σ ) along γ vanishing at the endpoints and with the property that∑N−1

i=0 ξt+iT = 0 for all t ∈ [0, T ], then the last part in the proof of Theorem 4.4 doesnot work. This suggests that nondegenericity of iterates should be dealt with by different(non variational) arguments, exactly as in the Riemannian case where the question ofiterate closed geodesics has been treated in [4] by dynamical techniques.

Finally, it is worth observing that an analogous degenericity result in the stationaryLorentzian manifold does not hold for geodesics of arbitrary causal character. In [8] anexplicit example of a degenerate timelike geodesic in a standard static manifold whosedegeneracy is preserved by arbitrary infinitesimal stationary perturbations of the metrictensor, is shown.

2. Notations and Preliminaries

2.1. Basic notations and references. Let M be a smooth Hausdorff paracompactmanifold with dim(M) ≥ 2 and let ∇ be an arbitrarily fixed symmetric connectionon T M . Given another (symmetric) connection ∇′ on T M , there exists a (symmetric)


(1, 2)-tensor � on M defined by:

∇′ = ∇ + �,

that will be called the Christoffel tensor of ∇′ relative to ∇. An affine connection ona manifold M induces a connection on every vector bundle constructed by functorialconstruction on T M , like for instance all tensor bundles T M∗(r) ⊗ T M (s). These factswill be used in the proof of Lemma 4.1.

Given a (semi-)Riemannian metric tensor g on M , we will denote by R the cur-vature tensor of the Levi–Civita connection of g, chosen with the sign convention:R(X, Y ) = [∇X ,∇Y ] − ∇[X,Y ]. Finsler metrics in general do not determine compati-ble symmetric connections. A more general construction is obtained by considering theChern connection associated to a Finsler metric, which is a connection defined on thevertical bundle over the tangent bundle of the Finsler manifold. In this paper we willconsider a very special class of Finsler metrics, called Randers metrics (see Sect. 3),that are associated to the choice of a Riemannian metric tensor g and a 1-form ω. Inthis situation, it will be more convenient for the purposes of the present paper to use theLevi–Civita connection of g rather than Chern’s connection.

Recommended references for all the background material in Lorentzian and semi-Riemannian geometry are the textbooks [7,20]. Specific Finsler geometry techniqueswill be used only marginally in this paper; the basic reference for this topic is the book[6]. An extensive study of the relation between light rays in stationary spacetimes andgeodesics in Randers metrics is carried out in the recent paper [9], which inspired partsof the present work.

2.2. Geodesics and Jacobi fields in stationary spacetimes. Let M be a differentiablemanifold and g a Lorentzian metric tensor on M ; (M, g) is said to be stationary ifit admits a timelike Killing vector field. A Lorentzian manifold (M, g) is said to bestandard stationary if M is given by a product M0 × R, where M0 is a differentiablemanifold, and the metric tensor g is of the form:

g(x,s) ((v, r), (v, r)) = gx (v, v) + gx (δ(x), v) r + gx (δ(x), v) r − β(x)rr , (2.1)

where x ∈ M0, s ∈ R, v, v ∈ Tx M0, r, r ∈ TsR ∼= R, g is a Riemannian metrictensor on M0, δ ∈ X(M0) is a smooth vector field on M0, and β : M0 → R+ isa smooth positive function on M0. The field Y = ∂s tangent to the lines {x0} × R,x0 ∈ M0, is a timelike Killing vector field in (M, g); an immediate computation showsthat g(x,s)(Y, Y ) = −β(x) for all (x, s) ∈ M0 ×R. We will endow (M, g) with the timeorientation defined by the timelike vector field; Y ; thus, a causal curve (x, s) in M isfuture pointing when g (δ(x), x) − β s is everywhere negative. Locally, every stationaryLorentzian metric tensor has the form (2.1). When the vector field δ in (2.1) vanishesidentically on M0, then the metric g is said to be standard static.

Let ∇ be the Levi–Civita connection of the metric g in T M0; given a smooth mapf0 : M0 → R, denote by ∇ f0 its gradient relative to the metric g and by H f0(x) :Tx M0 → Tx M0, x ∈ M0, the Hessian of f0 relative to g at the point x , which is thegx -symmetric linear operator on Tx M0 given by H f0(x)v = ∇v(∇ f0), for all v ∈ Tx M0.If x is a critical point of f0, then gx

(H f0(x)v,w

) = d2 f0(x)(v,w) is the standard sec-ond derivative of f0 at x . Let us also recall the notion of Hessian of the smooth vectorfield δ, which is the (1, 2)-tensor on M0:

Hess(δ)(v,w) = ∇v∇wδ − ∇∇vwδ,


where v,w ∈ T M0, and w is an arbitrary (local) extension of w to a vector field in M0.We will need the symmetric part of the Hessian of δ, defined by:

Hesss(δ)(v,w) = 12 [Hess(δ)(v,w) + Hess(δ)(w, v)] = Hess(δ)(v,w) − 1

2 R(v,w)δ,

for all v,w ∈ T M0, where R is the curvature tensor of ∇. It will be convenient tointroduce the following notation: by Hs(δ, v) : Tx M0 → M0 we will mean the linearoperator defined by:

Hs(δ, v)w = Hesss(δ)(v,w),

for all x ∈ M0 and all v,w ∈ Tx M0. Its g-adjoint, denoted by Hs(δ, v)�, is defined by:

g(Hs(δ, v)�w, z

) = g (Hs(δ, v)z, w) = g (Hesss(δ)(v, z), w),

for all x ∈ M0 and all v,w, z ∈ Tx M0. Moreover, the symbol Rs will denote the(1, 3)-tensor on M defined by Rs(a, b) : Tx M0 → Tx M0:

Rs(a, b)v = 12 (R(a, b)v + R(v, b)a),

for all x ∈ M0 and all a, b, v ∈ Tx M0. Its g-adjoint Rs(a, b)� is defined by:

g(Rs(a, b)�v,w

) = g (Rs(a, b)w, v)

for all x ∈ M0 and all a, b, v, w ∈ Tx M0. A curve γ (t) = (x(t), s(t)) in M is a geodesicrelative to the metric (2.1) if and only if its components x and s satisfy the system ofdifferential equations:

Ddt x + D

dt (s δ) − s (∇δ)�(x) + 12∇β(x) s2 = 0,

d

dt[g (δ(x), x) − β(x) s] = 0,

(2.2)

where Ddt denotes covariant differentiation along x relative to the connection∇, and (∇δ)�

is the (1, 1)-tensor on M defined by g ((∇δ)�(v), w) = g (∇wδ,w) for all v,w ∈ T M .Since Y is a Killing vector field, then for a geodesic γ = (x, s), the following quantityis constant:

cγ = g(γ , Y ) = g(δ, x) − β(x)s; (2.3)

if γ is a future pointing causal geodesic, then cγ < 0. The second variation of theg-geodesic action functional at a geodesic γ (t) = (x(t), s(t)), t ∈ [0, 1], is given by:

Ig,δ,β(γ )[(ξ, σ ), (ξ , σ )

]=∫ 1

0

[g

( Ddt ξ, D

dt ξ)

+ g(R(ξ, x)ξ , x

)+ s g

(Hesss(δ)(ξ, ξ ), x

)

+s g(∇ξ δ,

Ddt ξ

)+ s g

(∇ξ δ,

Ddt ξ

)+ σ ′ g

(∇ξ δ, x)

+ σ ′ g(∇ξ δ, x

)

+σ ′g( D

dt ξ, δ)

+ σ ′g( D

dt ξ , δ) − σ ′ s g (∇β(x), ξ) − σ ′ s g

(∇β(x), ξ)

+s g(R(ξ, x)ξ , δ

) − 12 s2 g

(Hβ(x)ξ, ξ

) − β(x)σ ′σ ′] dt ,

where ξ , ξ are variational vector fields along x vanishing at the endpoints, and σ, σ aresmooth functions on [0, 1] vanishing at 0 and at 1. In the above formula and in the rest ofthe section we will denote by a dot the derivatives of the components x and s of the curve


γ , and with a prime the derivatives of the component σ of the vector field V = (ξ, σ )

along γ . A pair V = (ξ, σ ) is a Jacobi field along the geodesic γ = (x, s) if it satisfiesthe second order linear system of differential equations:

D2

dt2 ξ − R(x, ξ) x − s Hs(δ, ξ)� x + Ddt

(s ∇ξ δ

) − s (∇δ)�( D

dt ξ) − σ ′ (∇δ)�(x)

+ Ddt (σ

′ δ) + σ ′ s ∇β(x) − s Rs(ξ, x)�δ + 12 s2 Hβ(x)ξ = 0, (2.4)

andddt

[g

(∇ξ δ, x)

+ g( D

dt ξ, δ) − s g (∇β(x), ξ) − β(x) σ ′] = 0. (2.5)

It is well known that lightlike geodesics and their conjugate points are invariant by con-formal changes of the metric (see for instance [19, Theorem 2.36]). Thus, it will notbe restrictive to assume in the remainder of the paper that β ≡ 1 in (2.1). Assume thatγ = (x, s) is a future pointing lightlike geodesic, so that:

g(x, x) + 2g(x, δ)s − s2 ≡ 0, s > g(δ, x), (2.6)

and thus:

s = g(x, δ) +√

g(x, x) + g(x, δ)2 = g(x, δ) − cγ . (2.7)

Substituting (2.7) into (2.2) (with β ≡ 1) gives the following differential equationsatisfied by the spatial part x of the lightlike geodesic γ = (x, s):

Ddt x + D

dt (g(x, δ) δ) − g(x, δ)(∇δ)� x + cγ

[(∇δ)� x − (∇δ)x

] = 0. (2.8)

Let V = (ξ, σ ) be a Jacobi field along γ ; from (2.4), (2.5) and (2.6), we obtain that ξ

satisfies the following integro-differential equation:

D2

dt2 ξ − R(x, ξ)x + (c0 − g(x, δ))[Hs(δ, ξ)� x + (∇δ)� D

dt ξ + Rs(x, δ)ξ]

− Ddt

[(c0 − g(x, δ)) ∇ξ δ

]+ D

dt

[(

A(x, ξ) −∫ 1

0A(x, ξ) dt

)

δ

]

−(

A(x, ξ) −∫ 1

0A(x, ξ) dt

)

(∇δ)� x = 0, (2.9)

where

A(x, ξ) = g( D

dt ξ, δ)

+ g(x,∇ξ δ

). (2.10)

2.3. An abstract genericity result for nondegenerate critical points. A subset of a metricspace is said to be generic if it contains the intersection of a countable family of denseopen subsets. Let us recall the following genericity result from [8,26]:

Proposition 2.1. Let B be a Banach manifold, X a Hilbert manifold, A ⊂ B × X anopen subset and F : A → R be a function of class Ck, with k ≥ 2. Set

Ab = {x ∈ H : (b, x) ∈ A},let Fb : Ab → R be defined by Fb(x) = F(b, x) for all x ∈ Ab and set:

C = {(b0, x0) ∈ A : x0 is a critical point of Fb0

}.


Assume that for all (b0, x0) ∈ C the following hypotheses are satisfied:

(a) the Hessian d2 Fb0(x0) = ∂2 F∂x2 (b0, x0) is Fredholm;

(b) for all ξ ∈ Ker(d2 Fb0(x0)

)with ξ �= 0 there exists β ∈ Tb0 B such that

∂2 F

∂b ∂x(b0, x0) [(β, ξ)] �= 0.

Then, denoting by : B × X → B the projection onto the first factor, the set ofb ∈ (A) such that Fb is a Morse function is generic in (A).

Proof. Condition (b) implies transversality of the map ∂ F∂x : B × X → T X ∗ to the zero

section of the cotangent bundle T X ∗, thus C is an embedded submanifold of the productB × X . Together with assumption (a), it also implies that the projection |C : C → Bis a nonlinear Fredholm operator of index 0, whose critical values are those b ∈ (A)

for which Fb is not a Morse function. The result is then obtained as an application of theSard–Smale theorem ([24]). See [26, Theorem 1.2] and [8, Sect. 3] for the details. �

3. Light Rays and Randers Geodesic

3.1. Randers metrics. Let M0 be a differentiable manifold. A Randers metric on M0is a pair (h, ω), where h is a Riemannian metric tensor on M0 and ω is a 1-form onM0 with ‖ωp‖ < 1 for all p ∈ M0. Here, ‖ · ‖ is the norm of linear forms inducedby h. Alternatively, a Randers metric can be described as a pair (h, X), where X is avector field on M0 with ‖X p‖ < 1 for all p ∈ M0, the relation between X and ω beingω = h(X, ·).

A Randers metric (h, ω) on M0 gives a Finsler metric f(h,ω) : T M0 → R obtainedby setting:

f(h,ω)(v) =√

h(v, v) + ω(v), v ∈ T M0.

Recall that a Finsler metric on M0 is a continuous function f : T M0 → [0, +∞[,smooth outside the zero section of T M0, with f (v) �= 0 for v �= 0, which is positivelyhomogeneous, and whose second derivatives in the directions of the vertical subbundleof T (T M0) are everywhere positive definite. A Finsler metric f defines a geometry onM0 in some aspects similar to the standard Riemannian geometry, but with interestingdifferences in several global properties. A Finsler geodesic is a smooth curve γ :[a, b] → M0 which (locally) minimizes its length L , defined by:

L(γ ) =∫ b

af (γ (t)) dt.

Finsler geodesics are also stationary points of the functional:

Q(γ ) =∫ b

af (γ (t))2 dt

defined in the space of paths γ joining two fixed endpoints. One should observe thatneither L nor Q are functionals of class C1 due to the lack of regularity of f . Station-ary points of Q are parameterized by constant Finsler speed, i.e., f (γ (t)) is constantalong γ .


For the special class of Randers metrics, however, one can study geodesics in termsof critical points of a smooth functional. Namely, given a Randers metric (h, ω), thefunctional:

F(h,ω)(γ ) =(∫ b

ah(γ , γ ) dt

) 12

+∫ b

aω(γ ) dt (3.1)

is smooth, and its critical points (in the space of fixed endpoints paths) are geodesics rel-ative to the Finsler metric f(h,ω) that are parameterized by constant Riemannian speed,i.e., h(γ , γ ) is constant along γ .

3.2. A first order Fermat principle. Let us now fix a Riemannian metric g and a smoothvector field δ on M0; the manifold M = M0 × R will be endowed with the stationaryLorentzian metric tensor (2.1) with β ≡ 1. Let p0, p1 ∈ M0 be fixed, and consider theHilbert manifold �p0,p1(M0) consisting of all curves x : [0, 1] → M0 of Sobolev classH1 and such that x(0) = p0, x(1) = p1. Consider the functional F : �p0,p1(M0) → Rdefined by:

F(x) =(∫ 1

0g(x, x) + g(x, δ)2 dt

) 12

+∫ 1

0g(x, δ) dt.

The reader will observe that F is the smooth functional (3.1) relative to the Randersmetric (h, ω) defined by:

h(v,w) = g(v,w) + g(δp, v)g(δp, w), ωp(v) = g(δp, v), (3.2)

for all p ∈ M0 and all v,w ∈ Tp M0.The Euler–Lagrange equation of F is given by:

Ddt x + D

dt (g(x, δ) δ) − g(x, δ)(∇δ)� x + cx[(∇δ)� x − (∇δ)x

] = 0 , (3.3)

where

cx (t) = −√

g(x, x) + g(x, δ)2. (3.4)

If x is a critical point of F , the quantity cx is constant on [0, 1], and the following equalityholds:∫ 1

0g

( Ddt ξ, x

)+ g(x, δ)

[g

( Ddt ξ, δ

)+ g

(x,∇ξ δ

)]dt = cx

∫ 1

0g

( Ddt ξ, δ

)+ g

(x,∇ξ δ

)dt

(3.5)

for all vector fields ξ along x with ξ(0) = ξ(1) = 0.

Remark. Observe that for every non constant critical point x of F the constant cx isstrictly negative, which in particular implies that x is an immersion, i.e., x never van-ishes. Thus, when p0 �= p1, every critical point of F is an immersion.

Comparing (3.3) with (2.8) gives the following result, originally proven in [10]:


Proposition 3.1. (First order Fermat principle) The map γ = (x, s) �→ x gives a bijec-tion from the set of future pointing lightlike geodesics in M = M0 × R from the event(p0, 0) and the observer U = {p1}×R to the set of geodesics in M0 from p0 to p1 relativeto the Randers metric (h, ω) given in (3.2). Its inverse is defined by x �→ γx = (x, sx ),where sx (t) = Gx (t) − cx t and Gx is the primitive of the function g(x, δ) satisfy-ing Gx (0) = 0. If x and γ are respectively a Randers geodesic and a lightlike geodesicrelated by the above bijection, then the constant cx corresponds to the constant cγ definedin (2.3). Moreover, the affine parameterization of γ corresponds to parameterization ofx with constant Riemannian speed.

Due to its geodesical nature, Eq. (3.3) shares many properties of a geodesic equation.For instance, the set of its solutions is invariant by affine reparameterizations, and thiswill be used systematically throughout (see for instance the proof of Lemma 4.3).

3.3. The second order Fermat principle. Let x ∈ �x0,x1(M0) be a critical point of Fand let cx be the constant (3.4); the second variation of F at x is obtained by a directcomputation, and it is given by:

d2 F(x) [ξ, η] = 1cx

(∫ 1

0g

(∇ξ δ, x)

+ g( D

dt ξ, δ)

dt

) (∫ 1

0g

(∇ηδ, x)

+ g( D

dt η, δ)

dt

)

− 1cx

∫ 1

0g

( Ddt ξ, D

dt η)+g (R(ξ, x)x, η)+

[g

( Ddt ξ, δ

)+g

(∇ξ δ, x)] [

g( D

dt η, δ)+g

(∇ηδ, x)]

dt

+ 1cx

∫ 1

0[cx − g(x, δ)]

[g (Hesss(δ)(ξ, η), x) + g

(∇ξ δ,Ddt η

)+ g

(∇ηδ,Ddt ξ

)

+g (Rs(ξ, x)η, δ)]

dt. (3.6)

Proposition 3.2. For all x ∈ �x0,x1(M0) critical point of F, the Hessian d2 F(x) isFredholm.

Proof. This is a standard argument. The first term in the second line of (3.6):

P[ξ, η] = − 1

cx

∫ 1

0g

( Ddt ξ, D

dt η)

dt

is represented by a positive isomorphism of Tx�x0,x1(M0) (recall: cx < 0). The dif-ference D[ξ, η] = d2 F(x) [ξ, η] − P[ξ, η] is represented by a compact operator ofTx�x0,x1(M0). Namely, the reader will observe that such bilinear form only containsat most one covariant derivative with respect to t of either ξ or η, i.e., each term ofD[ξ, η] is continuous with respect to the C0-topology in one of its variables, and theclaim follows from the fact that the inclusion H1 ↪→ C0 is compact. �

A straightforward computation using (3.6) shows that a vector field ξ ∈ Tx�x0,x1(M0)

is in the kernel of d2 F(x) if and only if ξ is of class C2; it vanishes at the endpointsξ(0) = ξ(1) = 0, and it satisfies the integro-differential equation (2.9). Recalling (2.10),we have a bijective correspondence between vectors ξ in the kernel of d2 F(x) and Jacobifields V = (ξ, σξ ) along the lightlike geodesic γ vanishing at the endpoints, where:

σξ (t) =∫ t

0A(x, ξ) dr − t ·

∫ 1

0A(x, ξ) dr.

We have therefore proven the following:


Proposition 3.3. (Second order Fermat principle) Let x ∈ �p0,p1(M0) be a criticalpoint of F, and let γ = (x, s) : [0, 1] → M = M0 × R be the corresponding futurepointing lightlike geodesic. Then, γ (1) is conjugate to γ (0) along γ if and only if x isa degenerate critical point of F. Moreover, the multiplicity of γ (1) as a conjugate pointalong γ equals the dimension of the kernel of d2 F(x).

The integro-differential equation (2.9) shares many properties of a standard Jacobi dif-ferential equation. For instance, all its solutions are defined globally. Moreover, the setof solutions of (2.9) along a periodic solution x of Eq. (3.3) with period T is invariantby T -translation. This fact will be used in formula (4.10).

4. Genericity of Lightlike Nondegeneracy

4.1. Tensors with arbitrarily prescribed value and covariant derivative along a curve.Let (M,∇) be an affine manifold, i.e., M is a differentiable manifold and ∇ is a con-nection in the tangent bundle T M . Given nonnegative integers r, s, we will denote byT M∗(r) ⊗ T M (s) the tensor product of r copies of T M∗ and s copies of T M ; sectionsof T M∗(r) ⊗ T M (s) are called tensors of type (s, r) on M . Let us show that one canconstruct globally defined tensors on M whose value and covariant derivative has beenprescribed on a sufficiently short piece of curve.

Lemma 4.1. Let (M,∇) be an affine manifold, let γ : [a, b] → M be a smooth immer-sion, and let V be a smooth vector field along γ . Let t0 ∈ ]a, b[ be an instant at which Vt0is not parallel to γ (t0). Then, there exists an open interval I ⊂ [a, b] containing t0 with

the property that, given any pair of smooth sections H, K ∈ �(γ ∗(T M∗(r) ⊗ T M (s))

)

over I , and given any open set U containing γ (I ), then there exists a globally defined(s, r)-tensor h on M with compact support contained in U, such that hγ (t) = Ht and∇Vt h = Kt for all t ∈ I . If both Ht and Kt are symmetric or anti-symmetric relativeto any of the variables for all t ∈ I , then the tensor h can be found to satisfy the samesymmetries or anti-symmetries relative to the same variables.

Proof. We choose an interval I containing t0 such that:

• γ is injective on I ;• Vt is never parallel to γ (t) for all t ∈ I ;• there exist:

– a frame e1, . . . , en of T M along γ |I , with e1(t) = γ (t) and e2(t) = Vt for allt ∈ I ;

– a positive number ε and an open subset W containing γ (I ) which is the counter-domain of a system of Fermi coordinates:

I ×]−ε, ε[n−1 (x1, x2, . . . , xn) �−→expγ (x1)(x2 · e2(x1) + · · · + xn · en(x1))∈W.

Obviously, the open set W can be chosen arbitrarily small, i.e., contained in any pre-scribed open set U containing γ (I ). We will now define the desired tensor h as follows.First, we denote by d the connection in T M |W which is trivial in the Fermi coordinates(x1, . . . , x2), and by �d the Christoffel tensor of d relative to ∇. Given any (s, r)-tensorh with support in W , then its covariant derivatives relative to the connections ∇ and d


are related by the identity:

dvh(α1, . . . , αr , v1, . . . , vs) = ∇vh(α1, . . . , αr , v1, . . . , vs)

+r∑

j=1

h(α1, . . . , �

dx (v)∗α j , . . . , αr , v1, . . . , vs

)

−s∑

k=1

h(α1, . . . , αr , v1, . . . , �

dx (v)vk, . . . , vs

),

for all x ∈ W , all α1, . . . , αr ∈ Tx M∗ and all v, v1, . . . , vs ∈ Tx M .

We will therefore define a smooth section K ∈ �(γ ∗(T M∗(r) ⊗ T M (s))

)over the

interval I by setting:

Kt (α1, . . . , αr , v1, . . . , vr ) = Kt (α1, . . . , αr , v1, . . . , vr )

−r∑

j=1

Ht

(α1, . . . , �

dγ (t)(Vt )

∗α j , . . . , αr , v1, . . . , vs

)

+s∑

k=1

Ht

(α1, . . . , αr , v1, . . . , �

dγ (t)(Vt )vk, . . . , vs

),

for all t ∈ I , all α1, . . . , αr ∈ Tγ (t)M∗ and all v1, . . . , vs ∈ Tγ (t)M .The problem is now to determine an (s, r)-tensor h with support contained in W ,

with hγ (t) = Ht and dVt h =(

d ∂∂x2

h

)

γ (t)= Kt for all t ∈ I . This is an easy task, using

coordinates (x1, . . . , xn), by setting:

h(x1, . . . , xn) = Hx1 + x2 · β(x2, . . . , xn)Kx1,

where β : W → R is a smooth nonnegative function with compact support, which isequal to one in a cylindrical neighborhood of I × 0n−1 (a segment on the x1-axis) inI × ]−ε, ε[n−1. Clearly, if both H and K have symmetries or anti-symmetries in any ofthe variables, then so does h. �

4.2. Banach spaces of tensors. In order to state our genericity result, we need tointroduce suitable Banach space structures on the set of tensors of class Ck , k ≥ 2,on the manifold M0. Since we are not assuming compactness for M0, there is no canon-ical choice of such structure.

We need two Banach spaces M and V satisfying the following axioms:

• Elements of M are symmetric (0, 2) tensors of class C2 on M0; M must contain alltensors of this type of class C∞ and have compact support.

• Elements of V are vector fields of class C2; V must contain all vector fields of classC∞ and have compact support.

• Convergence in the norm of M and of V must imply C2-convergence on compactsets.


Moreover, we will need an open subset A of M whose elements are everywhere positivedefinite (0, 2) tensors, i.e., Riemannian metrics on M0. For reference in the rest of thepaper, we will call a triple (M,V,A) as above an admissible triple of tensors for themanifold M0.

Examples of the admissible triple of tensors (M,V,A) can be constructed as fol-lows. Consider a fixed complete Riemannian metric g0 on M0, and let ∇0 be a fixedsymmetric connection on M0, for instance the Levi–Civita connection of g0. The metricg0 induces in a natural way a norm on each space (Tx M∗

0 )(r)⊗(Tx M0)(s), and ∇0 induces

a connection on every tensor bundle (T M∗0 )(r) ⊗ (T M0)

(s). Then, use these norms andconnections to define M as the set of (0, 2)-symmetric tensors h of class C2 on M0 suchthat:

c0(h) = supx∈M0

‖hx‖ < +∞, c1(h) = supx∈M0

‖∇0hx‖ < +∞,

c2(h) = supx∈M0

‖(∇0)2hx‖ < +∞,

and V the space of all vector fields V of class C2 on M0 such that:

d0(V ) = supx∈M0

‖Vx‖ < +∞, d1(V ) = supx∈M0

‖∇0Vx‖ < +∞,

d2(V ) = supx∈M0

‖(∇0)2Vx‖ < +∞.

A Banach space norm on M is given by:

‖h‖M = max {c0(h), c1(h), c2(h)} ,

and a Banach space norm on V is given by:

‖V ‖V = max {d0(V ), d1(V ), d2(V )} .

Completeness of the metrics ‖·‖M and ‖·‖V follows easily from the completeness of g.Clearly, M contains all symmetric (0, 2)-tensors of class C2 with compact support, andV contains all vector fields of class C2 with compact support. Moreover, convergencein these spaces implies C2-convergence on compact subsets of M0.

A typical open subset A of M consisting of everywhere positive definite tensors ingiven by:

A ={

h ∈ M : infx∈M0

λmin(hx ) > 0

}

,

where λmin(hx ) is defined by:

λmin(hx ) = minv∈Tx M0

gx (v,v)=1

hx (v, v).

Openness of A is proved easily using the fact that the map

T �−→ λmin(T ) = min〈v,v〉=1〈T v, v〉 ∈ R


is Lipschitz continuous1 on the space of all symmetric operators T on a finite dimensionalvector space with inner product 〈·, ·〉.

Given an admissible triple of tensors (M,V,A) for M0, then every element (g, δ) ∈ Adefines a stationary Lorentzian metric tensor gg,δ on M0 × R by:

gg,δ

(x,s) ((v, r), (v, r)) = gx (v, v) + gx (δ(x), v) r + gx (δ(x), v) r − rr . (4.1)

4.3. On the main result. We need a preliminary result concerning the distribution ofinstants along a lightlike geodesic at which the spatial component of a Jacobi field isparallel to the spatial component of the geodesic.

Lemma 4.2. Let M = M0 × R be a standard stationary Lorentzian manifold endowedwith the metric (2.1), let γ = (x, s) : [0, 1] → M0 × R be a lightlike geodesic, and letV = (ξ, σ ) be a Jacobi field along γ which is not everywhere parallel to γ . Then, theset:

T = {t ∈ [0, 1] : ξt is parallel to x(t)}is finite.

Proof. We will show that the set T coincides with the set:

{t ∈ [0, 1] : Vt is parallel to γ (t)}which is finite (see [8, Lemma 2.5]). Assume that at some instant t one has ξt = λx(t)for some λ ∈ R and write σ = λs +ρ. We will show that ρ = 0 as follows. First, observethat g(V, γ ) ≡ 0, because g(V, γ ) is an affine function vanishing at 0 and at 1. Thus,

λg (x(t), x(t)) + 2λg (x(t), δ) s + ρg (x(t), δ) − λs(t)2 − s(t)ρ = 0. (4.2)

Second, since γ is lightlike:

s = g(x, δ) ±√

g(x, x) + g(x, δ)2 = g(x, δ) ∓ cx . (4.3)

Substituting (4.3) into (4.2) gives:

ρ · cx = 0,

which yields ρ = 0. Here, we have cx �= 0, because γ = (x, s) is a lightlike geodesic,and thus x is not constant. �Note that Lemma 4.2 in particular applies when V is a nontrivial Jacobi field along γ

such that V (0) = V (1) = 0; such a vector field is not everywhere parallel to γ .As to the self-intersections of the spatial part of a lightlike geodesic, we have the

following:

Lemma 4.3. Let M = M0 × R be a standard stationary Lorentzian manifold endowedwith the metric (2.1), let γ = (x, s) : [0, 1] → M0 × R be a lightlike geodesic.Then, either x is a periodic curve with period T < 1, or x has only a finite number ofself-intersections.

1 The following inequality holds for all symmetric operators S and T : |λmin(T ) − λmin(S)| ≤ ‖T − S‖.


Proof. Assume the existence of sequences (rn)n and (qn)n in [0, 1] with rn �= qn andx(qn) = x(rn) for all n ∈ N. We can also assume that the limits lim rn = r andlim qn = q exist and, since x is an immersion and thus locally injective, that rn �= r ,qn �= q for all n; clearly, x(q) = x(r). Moreover, by the local injectivity, r �= q, sayr > q. For t near q, set y(t) = x(t + r − q); this is the spatial part of some light-like geodesic in M with constant cy equal to cx . It is y(q) = x(r) = x(q); moreover,q ′

n = rn − r + q is a sequence tending to q with y(q ′n) = x(rn) = x(qn) for all n. This

implies that the tangent vectors y(q) and x(q) are linearly dependent; since cy = cx ,one deduces immediately that y(q) = ±x(q), hence x(r) = ±x(q). We claim thatx(r) = x(q). For, assume by contradiction x(r) = −x(q); set w(t) = x(r − t) fort ∈ [q, r ]. Since w(r − q) = x(q) = x(r) and w(r − q) = −x(q) = x(r), by unique-ness we have x(r + q − t) = w(t −q) = x(t) for all t ∈ [q, r ]; computing the derivativeat t = 1

2 (r + q) we have:

−x( r+q

2

) = x( r+q

2

) �⇒ x( r+q

2

) = 0,

which is absurd, and proves that x(r) = x(q). Again, a uniqueness argument shows thatx has a periodic extension to R with period T = r − q; we need to show that T < 1.Assume by contradiction T = 1, i.e., q = 0 and r = 1. Consider x extended to thewhole real line by periodicity; we have x(rn − 1) = x(rn) = x(qn) for all n ∈ N, whichcontradicts the local injectivity of x at 0. This concludes the proof. �We will now set ourself the task of proving our main result on the genericity of thenondegeneracy condition for light rays. To this aim, we will use Proposition 2.1, whoseassumption (b) involves the computation of a second mixed partial derivative, which isperformed as follows. Let (M,V,A) be an admissible triple of tensors for M0 as definedin Sect. 4.2. Set g = (g, δ) ∈ A×V and consider the functional F : A×V×�p0,p1(M0):

F(g, x) =(∫ 1

0g(x, x) + g(x, δ)2 dt

) 12

+∫ 1

0g(x, δ) dt. (4.4)

Let g0 = (g0, δ0) ∈ A × V be fixed, and assume that x0 ∈ �p0,p1(M0) is a criti-cal point of F(g0, ·). Let h = (h, ζ ) ∈ Tg0A × Tδ0V

∼= M × V be an infinitesimalvariation of g0 and ξ ∈ Tx0�p0,p1(M0) an infinitesimal variation of x0. Then, setting

c0 = −√

g0(x0, x0) + g0(x0, δ0)2, the second mixed derivative ∂2 F∂g ∂x (x0, g0) [h, ξ ] is

computed as:

∂2 F

∂g ∂x(g0, x0) [h, ξ ]

= 1

c30

(∫ 1

0g0

(x0,

Ddt ξ

)+ g0(x0, δ0)

[g0

( Ddt ξ, δ0

)+ g0

(x0,∇ξ δ0

)]dt

)

·(∫ 1

0h(x0, x0) + 2g0(x0, δ0) [h(x0, δ0) + g0(x0, ζ )] dt

)

(4.5)

− 1

2c0

∫ 1

0∇ξh(x0, x0) + 2h

( Ddt ξ, x0

)

+ 2[g0

( Ddt ξ, δ0

)+ g0

(x0,∇ξ δ0

)][h(x0, δ0) + g0(x0, ζ )] dt


− 1

2c0

∫ 1

02g0(x0, δ0)

[∇ξh(x0, δ0) + h( D

dt ξ, δ0)

+ h(x0,∇ξ δ0

)+ g0

( Ddt ξ, ζ

)+ g0

(x0,∇ξ ζ

)]dt

+∫ 1

0∇ξh(x0, δ0) + h

( Ddt ξ, δ0

)+ h

(x0,∇ξ δ0

)+ g0

( Ddt ξ, ζ

)+ g0

(x0,∇ξ ζ

)dt.

Setting ζ = ∇ξ ζ ≡ 0 and h ≡ 0 in the right hand side of (4.5), we get:

∂2 F

∂g ∂x(g0, x0) [h, ξ ] = 1

2c0

∫ 1

0∇ξh (x0, 2 [c0 − g0(x0, δ0)] δ0 − x0) dt; (4.6)

while setting ζ ≡ 0 and h = ∇ξh ≡ 0 in the right hand side of (4.5), we get:

∂2 F

∂g ∂x(g0, x0) [h, ξ ] = 1

c0

∫ 1

0[c0 − g0(x0, δ0)] g0

(x0,∇ξ ζ

)dt. (4.7)

Note that the quantity c0 − g0(x0, δ0) is strictly negative on [0, 1]. It should also beobserved that the vector field W = 2 [c0 − g0(x0, δ0)] δ0 − x0 along x0 is never zero.Namely, if it were W = 0 at some instant, then at this instant it would be δ0 = λ · x0 forsome (negative) real number λ, and therefore:

0 = g0(W, x0) = 2

[

−√

g0(x0, x0)(1 + λ2) − λ2g0(x0, x0)

]

− g0(x0, x0).

However, since x0 is an immersion (Remark 3.2), the right hand side of the above formulais strictly negative, thus W never vanishes.

We are finally ready for the statement and the proof of our main result.

Theorem 4.4. Let M0 be a differentiable manifold, let p0, p1 ∈ M0 be distinct pointsand let (M,V,A) be an admissible triple of tensors for M0. Then, the set of all (g, δ) ∈ Asuch that the stationary Lorentzian metric gg,δ in M = M0 ×R defined in (4.1) has onlynondegenerate light rays from (p0, 0) to the observer U = {(p1, s) : s ∈ R} is genericin A. More precisely, the following stronger result holds. For δ ∈ V, denote by Mδ ⊂ Mthe open set:

Mδ = {g ∈ M : (g, δ) ∈ A},

and for g ∈ M, let Vg ⊂ V be the open set:

Vg = {δ ∈ V : ∃ g ∈ M with (g, δ) ∈ A}.

(1) For fixed g ∈ M, the set of δ ∈ Vg such that the metric tensor gg,δ definedin (4.1) has only nondegenerate light rays from the event (p0, 0) to the observerU = {(p1, s) : s ∈ R} is generic in Vg.

(2) For fixed δ ∈ V, the set of g ∈ Mδ such that the metric tensor gg,δ definedin (4.1) has only nondegenerate light rays from the event (p0, 0) to the observerU = {(p1, s) : s ∈ R} is generic in Mδ .


Proof. The result will be proven as an application of Proposition 2.1, considering thefunction F : A × �p0,p1(M0) → R defined by (4.4). By Proposition 3.3, nondegener-acy of all light rays from (p0, 0) to U for the metric tensor gg,δ is equivalent to the factthat the functional x �→ F

(gg,δ, x

)is Morse. Let g0 = (g0, δ0) ∈ A be fixed, and let

x0 ∈ �p0,p1(M0) be a critical point of the functional x �→ F(g0, x). The fact that the

second derivative ∂2 F∂x2 (g0, x0) is Fredholm, which is assumption (a) in Proposition 2.1,

is proved in Proposition 3.2.As to assumption (b) in Proposition 2.1, in our geodesic setup this is translated into

the following condition: given g0 = (g0, δ0) ∈ A, a critical point x0 ∈ �p0,p1(M0)

of the functional x �→ F(g0, x) and a non-trivial Jacobi field V = (ξ, σ ) along thecorresponding lightlike geodesic γ = (x, s), with V (0) = V (1) = 0, then there shouldexist an element h = (h, ζ ) ∈ Tg0A × Tδ0V

∼= M × V such that:

∂2 F

∂g ∂x(g0, x0) [h, ξ ] �= 0. (4.8)

More precisely, in order to prove statement (1) in the thesis, we need to show thatinequality (4.8) holds for some choice of h of the form h = (0, ζ ) (i.e., h = 0), whilestatement (2) will be proved by showing that (4.8) will hold for some choice of h of theform h = (h, 0) (i.e., ζ = 0). With such choices of h, we will have to show that theright-hand side of formulas (4.6) and (4.7) are not zero.

Since M (resp., V) contains all symmetric (0, 2)-tensors (resp., all vector fields)having compact support in M0, it will suffice to search for smooth tensors h and vectorfields ζ with the desired property and having compact support. Let us fix g0, x0 andV = (ξ, σ ) as above; by the assumption that p0 �= p1, we have that x0 is not constant,and thus it is an immersion (Remark 3.2). We will distinguish two cases: when x0 is notperiodic of period T < 1 and when x0 has period T < 1. Assume x0 not periodic ofperiod T < 1. By Lemma 4.3, x0 has only a finite number of self-intersections. UsingLemma 4.2, we can find a non-empty open subset ]a, b[ ⊂ [0, 1] and an open subset Uof M0 with the following properties:

(i) x−10 (U ) = ]a, b[;

(ii) x0|]a,b[ is an embedding;(iii) ξt is not parallel to x0(t) for all t ∈ ]a, b[.As we have already observed, the vector field W = 2 [c0 − g0(x0, δ0)] δ0 − x0 alongx0 never vanishes, and thus there exists a smooth field K of symmetric bilinear formsalong x0|[a,b] having compact support in ]a, b[ such that:

∫ b

aKt (x0, 2 [c0 − g0(x0, δ0)] δ0 − x0) dt �= 0.

Using (ii) and (iii), Lemma 4.1 tells us that there exists a globally defined smooth (0, 2)-symmetric tensor h on M0 having compact support contained in U such that hx0(t) = 0and ∇ξt h = Kt for all t ∈ ]a, b[. Then:

1

2c0

∫ 1

0∇ξh (x0, 2 [c0 − g0(x0, δ0)] δ0 − x0) dt

by (i)= 1

2c0

∫ b

a∇ξh (x0, 2 [c0 − g0(x0, δ0)] δ0 − x0) dt

= 1

2c0

∫ b

aKt (x0, 2 [c0 − g0(x0, δ0)] δ0 − x0) dt �= 0.


Similarly, since [c0 − g0(x0, δ0)] x0 never vanishes, there exists a smooth vector fieldK along x0|[a,b] having compact support in ]a, b[ such that:

∫ b

a[c0 − g0(x0, δ0)] g0 (x0, K ) dt �= 0.

Another application of Lemma 4.1 gives us the existence of a globally defined smoothvector field ζ on M0 having compact support contained in U such that ζx0(t) = 0 and∇ξt ζ = Kt for all t ∈ [a, b]. Thus:

1

c0

∫ 1

0[c0 − g0(x0, δ0)] g0

(x0,∇ξ ζ

)dt

by (i)= 1

c0

∫ b

a[c0 − g0(x0, δ0)] g0

(x0,∇ξ ζ

)dt

1

c0

∫ b

a[c0 − g0(x0, δ0)] g0 (x0, K ) dt �= 0.

This concludes the proof in the case that x0 is not periodic of period T < 1. Assume nowthat x0 is periodic, of period T < 1; for this case the proof goes along the same linesas the proof of [8, Prop. 4.3], that will be repeated here for the reader’s convenience.Consider the following numbers:

t∗ = min {t > 0 : x0(t) = q} , k∗ = max {k ∈ Z : kT < 1},for which the following inequalities hold:

∗ ≥ 1, 0 < t∗ < T, 1 = k∗T + t∗.

The curves x1 = x0|[0,t∗] and x2 = x0|[t∗,T ] join p1 and p2 (x2 with the opposite orien-tation), and the first part of the proof applies to both x1 and x2. Thus, we can find openintervals I1 = [a1, b1] ⊂ [0, t∗] and I2 = [a2, b2] ⊂ [t∗, T ] such that:

(iv) t ∈ I1, s ∈ ([0, t∗]\I1) ∪ [t∗, T ] implies x0(s) �= x0(t);(v) t ∈ I2, s ∈ ([t∗, T ]\I2) ∪ [0, t∗] implies x0(s) �= x0(t).

We can also find open subsets U1, U2 ⊂ M , with x0(Ii ) ⊂ Ui , i = 1, 2, satisfying:

x0(t) ∈ U1 ∩ x0(I1) ⇐⇒ ∃ r ∈ {0, . . . , k∗} such that t − rT ∈ I1,

x0(t) ∈ U2 ∩ x0(I2) ⇐⇒ ∃ r ∈ {0, . . . , k∗ − 1} such that t − rT ∈ I2. (4.9)

For j = 1, 2, consider the vector field η j along x j defined by:

η1t =

k∗∑

r=0

ξt+rT , η2t =

k∗−1∑

r=0

ξt+rT . (4.10)

Note that η1 and η2 are the spatial components of Jacobi fields V 1 and V 2 along alightlike geodesics γ0 = (x0, s0) in the stationary manifold M = M0 × R, and thusLemma 4.2 applies in this situation. It is not the case that both η1 and η2 are everywhereparallel to x0 on I1 and I2 respectively, for otherwise from (4.10) one would concludeeasily that ξ would be everywhere parallel to x0, and this is not the case since V = (ξ, σ )

is a non trivial Jacobi field vanishing at 0 and 1 (Lemma 4.2). Assume that, say, η1 is not


everywhere parallel to x0 on I1, i.e., by Lemma 4.2, there are only isolated values of t ,where η1

t is parallel to x0(t); the other case is totally analogous. By reducing the size ofI1, we can assume that η1

t is never a multiple of x0(t) on I1. Now, the first part of the proofcan be repeated, by replacing the vector field ξ with η1, as follows. Using Lemma 4.1,we can find a globally defined symmetric (0, 2)-tensor h on M0, with compact supportcontained in U1, with prescribed value hx0(t) = 0 for t ∈ I1, and covariant derivative∇η1

th = Kt along the curve x0|I1 such that:

∫ b1

a1

Kt (x0, 2 [c0 − g0(x0, δ0)] δ0 − x0) dt �= 0.

The choice of such K is possible, due to the fact that both vectors [c0 − g0(x0, δ0)] δ0−x0and x0 never vanish. For such a tensor h, we compute:

1

2c0

∫ 1

0∇ξh (x0, 2 [c0 − g0(x0, δ0)] δ0 − x0) dt

by (4.9)= 1

2c0

k∗∑

r=0

∫ b1+rT

a1+rT∇ξh (x0, 2 [c0 − g0(x0, δ0)] δ0 − x0) dt

= 1

2c0

k∗∑

r=0

∫ b1

a1

∇ξt+rT h (x0, 2 [c0 − g0(x0, δ0)] δ0 − x0) dt

= 1

2c0

∫ b1

a1

∇η1th (x0, 2 [c0 − g0(x0, δ0)] δ0 − x0) dt

= 1

2c0

∫ b1

a1

Kt (x0, 2 [c0 − g0(x0, δ0)] δ0 − x0) dt �= 0.

Similarly, by Lemma 4.1 one can find a globally defined vector field ζ on M0, havingcompact support contained in U1, with ζx0(t) = 0 and with prescribed values ∇η1

tζ = Kt

for all t ∈ I1, where:

∫ b1

a1

[c0 − g0(x0, δ0)] g0 (x0, Kt ) dt �= 0.

The choice of such K is possible, due to the fact that x0 never vanishes. For such a vectorfield ζ , one computes:

1

c0

∫ 1

0[c0 − g0(x0, δ0)] g0

(x0,∇ξ ζ

)dt

by (4.9)= 1

c0

k∗∑

r=0

∫ b1+rT

a1+rT[c0 − g0(x0, δ0)] g0

(x0,∇ξ ζ

)dt

= 1

c0

k∗∑

r=0

∫ b1

a1

[c0 − g0(x0, δ0)] g0(x0,∇ξt+rT ζ

)dt


= 1

c0

∫ b1

a1

[c0 − g0(x0, δ0)] g0

(x0,∇η1

tζ)

dt

= 1

c0

∫ b1

a1

[c0 − g0(x0, δ0)] g0 (x0, Kt ) dt �= 0.

The case in which η2 is not everywhere parallel to x0 is analogous, and this concludesthe proof. �A similar genericity result may be proved also in specific classes of stationary Lorentzianmetric tensors, for instance the class of globally hyperbolic tensors. This can be donealong the lines of an analogous result proved in [8, Sect. 4.5].

Acknowledgement. The authors thankfully acknowledge the help provided by E. Caponio, M. A. Javaloyesand A. Masiello during uncountable conversations.

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Communicated by G. W. Gibbons

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