Nonlinear Analysis 51 (2002) 607 – 631
www.elsevier.com/locate/na
Periodic trajectories in G!odel type space–times
A.M. Candelaa; ∗;1 , A. Salvatorea , M. S'anchezb; 2
a Dipartimento
Interuniversitario di Matematica, Universita degli Studi di Bari, Via E. Orabona 4,
70125 Bari, Italy
b Departamento de Geometr&'a y Topolog&'a, Facultad de Ciencias, Universidad de Granada,
Avenida Fuentenueva s=n, 18071 Granada, Spain
Received 15 April 2001; accepted 11 May 2001
Keywords: G!odel type space–times; (Y; T )-periodic trajectories; Closed geodesics; Ljusternik–
Schnirelman theory
1. Introduction
Aim of this paper is to prove the existence of periodic trajectories in a special class
of Lorentzian manifolds, the so-called G!odel type space–times.
Denition 1.1. A semi-Riemannian manifold (M; ·; ·L ) is a Lorentzian manifold of
G!odel type if there exists a smooth connected 4nite dimensional Riemannian manifold
(M0 ; ·; ·R ) such that M = M0 × R2 and ·; ·L is such that
·; ·L = ·; ·R + A(x) dy2 + 2B(x) dy dt − C(x) dt 2 ;
(1.1)
where x ∈ M0 , the variables (y; t) are the natural coordinates of R2 and A, B, C are
C 1 scalar 4elds on M0 such that
H(x) = B2 (x) + A(x)C(x) ¿ 0
for all x ∈ M0 :
(1.2)
(Of course, we are making some natural identi4cations with the product structure;
hence, it is Tz M ≡ Tx M0 × R2 for any z = (x; y; t) ∈ M).
∗ Corresponding author. Fax: 0039-0805963612.
E-mail address: [email protected] (A.M. Candela).
1 Supported by M.U.R.S.T. (research funds ex 40% and 60%).
2 Partially supported by a DGICYT Grant PB97-0784-C03-01.
c 2002 Elsevier Science Ltd. All rights reserved.
0362-546X/02/$ - see front matter PII: S 0 3 6 2 - 5 4 6 X ( 0 1 ) 0 0 8 4 6 - X
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A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
Remark 1.2. The assumption (1.2) implies that the metric de4ned in (1.1) is Lorentzian.
In fact, the number of the negative eigenvalues of ·; ·L is independent of z ∈ M and
equal to 1 (for more details on semi-Riemannian manifolds, cf. [2,16]). Notice that, under our de4nition, a G!odel type space–time may even not be time-orientable: take M0
equal to a circle S 1 and put, for each x = ei ∈ S 1 , A(x) = C(x) = cos ; B(x) = sin . It
is not diKcult to check that the lightcones rotate after a round to S 1 , and, thus, this
G!odel type space–time is not time-orientable.
Let us point out that De4nition 1.1 has been introduced in order to generalize the
classical G!odel Universe, which is an exact solution of the Einstein’s 4eld equations
in which the matter takes the form of a rotating pressure–free perfect Muid obtained
by Kurt G!odel in 1949 (cf. [12]). Anyway, such a de4nition covers also other models
of Lorentzian manifolds interesting from a physical point of view, for example the
G!odel–Synge space–times (cf. [19]), some types of Kerr–Schild manifolds and other
examples already introduced in [8] (see also [9] and references therein). From a Physics
point of view, the importance of this type of space–times justi4es the study of their
geometrical properties and, in particular, of their geodesics. On the other hand, from
a purely mathematical viewpoint, it is worth mentioning the following technical point.
Variational methods have been systematically used in order to study some properties of
many space–times, as geodesic connectedness or the existence of periodic trajectories
(see, for example, [4,14]). In particular, more accurate results have been obtained
if, in addition, the space–time is (standard) static or stationary, i.e. it is a product
manifold M0 × R endowed with a Lorentzian metric which is spacelike on the vectors
tangent to the slices M0 × {t0 }; t0 ∈ R, and is independent of the time variable t on
R, so that the vector 4eld @t is timelike and Killing (in the static case there are no
cross terms between the space part M0 and the time axis R, which may exist in the
stationary case). Indeed, in this case the problem can be reduced to a “Riemannian”
one. Furthermore, extensions of these results can be obtained when one considers a
semi-Riemannian manifold of index and a distribution of independent vector 4elds
K1 ; : : : K which commute and span a maximal negative de4nite subspace at each point
(cf. [11]). G!odel type space–times admit two Killing vector 4elds, @y and @t , but none
of them is necessarily timelike; in fact, their causal characters may even change on
the manifold. In spite of this, it was shown by some of the authors in [8] that a
reduction to a Riemannian problem is still possible, and we will take advantage of
this fact.
Recall that if (M; ·; ·L ) is a manifold of G!odel type, or more in general a semiRiemannian manifold, a smooth curve z : [a; b] → M (b ¿ a) is a geodesic in M if
Ds z(s)
˙ =0
for all s ∈ [a; b];
where z˙ is the tangent 4eld along z and Ds z˙ is the covariant derivative of z˙ along z
induced by the Levi–Civita connection of ·; ·L . Furthermore, if z : [a; b] → M is a
geodesic, there exists a constant Ez , named energy of z, such that
Ez ≡ z(s);
˙
z(s)
˙
L
for all s ∈ [a; b]:
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609
Hence, z is said timelike, respectively, lightlike, spacelike, if its energy Ez is negative,
respectively null, positive. This classi4cation is called causal character of geodesics
and comes from General Relativity (cf. [16]).
Particular geodesics in G!odel type manifolds are the (Y; T )-periodic trajectories which
we de4ne as follows.
Denition 1.3. Let Y , T ∈ R be 4xed. A geodesic z : [a; b]→M, z(s) = (x(s); y(s); t(s)),
is a (Y; T )-periodic trajectory in M if the following conditions hold

ẋ(b) = ẋ(a);

 x(b) = x(a);

y(b) = y(a) + Y;
ẏ(b) = ẏ(a);



t(b) = t(a) + T;
t˙(b) = t˙(a):
(1.3)
In particular, a closed geodesic is a (0; 0)-periodic trajectory.
As the metric (1.1) is independent of the variables y and t, we choose the normalization y(a) = t(a) = 0 in order to avoid trivial multiplicity results. Moreover, without
loss of generality, in what follows we assume [a; b] = [0; 1].
Remark 1.4. When it is C ¿ 0, the physical interpretation of (0; T )-periodic trajectories in G!odel type space–times is equal to the physical interpretation of T -periodic
trajectories in standard stationary space–times: each observer travelling through the integral curves of @t may interpret that each slice with constant variable t is his “space”
(at least if A ¿ 0), and each (0; T )-periodic trajectory is truly periodic for him in his
space (analogously, this happens when A ¡ 0 for (Y; 0)-periodic trajectories). Moreover, for each (Y; T ) = (0; 0) a new couple of variables (u; v) ≡ (u(y; t); v(y; t)) can be
de4ned by means of the linear relations y = Yv − Tu, t = Tv + Yu and a (Y; T )-periodic
trajectory would be also periodic in u and would have periodic derivative in v with
v(1) − v(0) = 1. Thus, if @v = Y@y + T@t is timelike, then (Y; T )-periodic trajectories can
be seen as 1-periodic trajectories of a stationary space–time. Other interpretations are
also possible for other kinds of reference frames (in the sense of [20]) making natural
for the trajectories the double control of the advanced steps Y and T .
Remark 1.5. Let Y , T ∈ R. A (Y; T )-periodic trajectory zU : [0; 1] → M, z(s)
U = (x(s);
U
y(s);
U
tU(s)), will be called trivial if there exists xU ∈ M0 such that x(s)
U ≡ xU is a constant
curve. It is not diKcult to check from the geodesic equations that a curve z(s)
U
is
a trivial (Y; T )-periodic trajectory, if and only if x(s)
U
is a constant curve x(s)
U
≡ x,
U
where xU is a critical point of Y 2 A(x) + 2YTB(x) − T 2 C(x), and y,
U tU are straight lines of
slope Y; T , respectively (see also Remark 3.4). Clearly, for any couple (Y; T ), trivial
(Y; T )-periodic trajectories always occur when M0 is compact.
In this paper a variational approach allows to prove that, under suitable assumptions
on the coeKcients A, B, C, for any non-zero couple (Y; T ) ∈ R2 there exist in4nitely
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many non-trivial spacelike (Y; T )-periodic trajectories in M with higher and higher
energy.
Previous results on periodic trajectories are stated in manifolds where just one of
the periods, say T , can be taken into account. Results in Lorentzian manifolds of
splitting type were obtained with variational methods in [4] (static case, i.e., an orthogonal splitting metric independent of time), [5] (static case with boundary), [6] (orthogonal splitting metrics depending on time) and [1] (stationary case). Moreover, other
results for non-orthogonal splitting metrics depending on time have been obtained with
non-variational methods in [21] (see also references therein).
Recall that all the previous manifolds (but those ones with boundary) are globally
hyperbolic and the results obtained with non-variational methods are applicable just to
causal (i.e., timelike or lightlike) trajectories.
Let us point out that a G!odel type space–time with A; C ¿ 0 is a stationary manifold
and, in particular, if B ≡ 0 the space–time is static. Then, some results for the static
case can be obtained under our approach. Nevertheless, the previous restrictions will
not be imposed and also the study of spacelike geodesics will be covered.
Here, we use a variational principle which is essentially the same introduced in [8]
in order to study the geodesical connectedness in G!odel type space–times. However,
with respect to the case of geodesics joining two points, two more problems arise in the
study of periodic trajectories: the existence of trivial solutions (hence, it is interesting
to look for non-trivial ones) and the lack of compactness if M0 is just complete (then,
we need some controls at in4nity).
We will use the following notations:
1
1
1
A(x)
B(x)
C(x)
a(x) =
ds;
b(x) =
ds;
c(x) =
ds;
(1.4)
H(x)
H(x)
H(x)
0
0
0
L(x) = b2 (x) + a(x)c(x)
(1.5)
for any continuous curve x : [0; 1] → M0 ; moreover, 1 (M0 ) will denote the space of
closed H 1 -curves in M0 (see Section 2 for the precise de4nition).
We can state the main theorems.
Theorem 1.6. Let (M; ·; ·L ) be a Lorentzian manifold of G;odel type. Let us assume
that
(H1 ) (M0 ; ·; ·R ) is a compact Riemannian manifold such that its fundamental group
1 (M0 ) is >nite or it has in>nitely many conjugacy classes;
(H2 ) for all x ∈ 1 (M0 ) it is |L(x)| ¿ 0;
(H3 ) there exist some positive constants k1 ; k2 ; k3 such that for all x ∈ 1 (M0 ) it is
a(x) b(x) c(x) 6 k1 ;
6 k2 ;
L(x) L(x) L(x) 6 k3 :
Then; for any non-zero couple (Y; T ) ∈ R2 there exist in>nitely many non-trivial spacelike distinct (Y; T )-periodic trajectories in M whose energies diverge positively.
A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
611
It is possible to extend Theorem 1.6 to G!odel type space–times M = M0 × R2 when
the Riemannian part M0 is just complete. To this aim we add some assumptions on
the metric (see (H4 )) and, as in [5] or [14], we introduce a suitable function U which
is strictly convex at in4nity (see (H5 ) and (H6 )).
Theorem 1.7. Let (M; ·; ·L ) be a G;odel type space–time such that
(H1 ) (M0 ; ·; ·R ) is complete; not contractible in itself and its fundamental group
1 (M0 ) is >nite or it has in>nitely many conjugacy classes.
Let us assume that (H2 ); (H3 ) hold and; moreover;
(H4 ) there exist some positive constants K1 ; K2 ; K3 such that for all x ∈ M0 the
coeBcients A; B and C satisfy
A(x) B(x) C(x) 6 K1 ;
6 K2 ;
H(x) H(x) H(x) 6 K3 ;
(H5 ) there exist x0 ∈ M0 ; U ∈ C 2 (M0 ; R+ ) and some positive constants R; #; $; such
that
x ∈ M0 ;
d(x; x0 ) ¿ R ⇒ HRU (x)[&; &] ¿ $&; &R
for all & ∈ Tx M0 ;
(H6 ) taken x0 and U as in (H5 ); there results
lim inf
∇U (x); ∇A(x)R ¿ 0;
lim
∇U (x); ∇B(x)R = 0;
d(x; x0 )→+∞
d(x; x0 )→+∞
lim sup ∇U (x); ∇C(x)R 6 0;
d(x; x0 )→+∞
where d(·; ·) is the distance in M0 and HRU (x)[&; &] denotes the Hessian of the function
U at x in the direction & both induced on M0 by its Riemannian structure ·; ·R .
Then; for any non-zero couple (Y; T ) ∈ R2 there exist in>nitely many non-trivial spacelike distinct (Y; T )-periodic trajectories in M whose energies diverge positively.
Remark 1.8. When the hypothesis (H1 ) (respectively (H1 )) on 1 (M0 ) does not hold
in Theorem 1.6 (respectively in Theorem 1.7) the existence of at least one non-trivial
(Y; T )-periodic trajectory can also be stated (see Remarks 3.7, 4.3).
Let us point out that the hypotheses in Theorems 1.6 and 1.7 can be simpli4ed
if Y = 0 or T = 0. Indeed, if Y = 0 (respectively, T = 0) assumptions (H3 ) and (H4 )
become
a(x) c(x) 6 k3
6 k1
(H3 ) resp:
for all x ∈ 1 (M0 );
L(x) L(x) A(x) C(x) 6 K1
6
K
(H4 ) resp:
for all x ∈ M0 :
3
H(x) H(x) 612
A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
In particular, if we look for (0; 0)-periodic trajectories, i.e., closed geodesics in M,
also the hypotheses (H3 ) and (H4 ) can be avoided. In fact, it can be proved that if
(H2 ) holds, then z = (x; y; t) is a closed geodesic in M, if and only if x is a closed
geodesic in the Riemannian manifold M0 while y, t are constant (cf. Remark 2.2; the
hypothesis (H2 ) is unavoidable, see Section 5(C)).
Anyway, Theorems 1.6 and 1.7 do not imply the existence of in4nitely many (0,0)periodic trajectories, and so, in order to look for closed geodesics in a G!odel type
space–time, it is enough to study the existence of closed geodesics in a Riemannian
manifold; hence, all the known results in this 4eld can be applied (see, e.g., [13] for
classical results and [7] and its references for some more recent ones).
Remark 1.9. In [8, Remark 1.4] simple conditions on A; B; C which imply (H2 ); (H3 )
are stated. However, recall that if M0 is compact and (H2 ) is replaced by the stronger
condition
(H2 ) there exists ¿ 0 such that |L(x)| ¿ for all x ∈ 1 (M0 ),
then (H3 ) holds.
On the other hand, it is also easy to give conditions on A; B; C in order to obtain
timelike (Y; T )-periodic trajectories, even though they must be more restrictive in order
to avoid a too bad behavior of lightcones. For example, assume that M0 is compact,
A; C ¿ 0 and |B| ¿ ' ¿ 0; then, for any 4xed Y = 0 with sign Y = − sign B the number
of (Y; T )-periodic timelike trajectories goes to in4nity when T → ∞. In particular, this
always happens in the warped case A ≡ C ≡ 0, studied in Section 5(B).
In Section 5, we will give some examples where Theorems 1.6 and 1.7 can be
applied, and discuss their hypotheses further. More precisely, we apply these theorems:
4rst, to a simple example where the causal character of @y , @t changes and, second, to
the warped case. We also discuss how results for static space–times can be obtained and
give a new result, Corollary 5.1. Finally, the classical G!odel space–time is also studied.
We point out that previous theorems do not apply to this manifold because it does not
satisfy their assumptions. In fact, it does not have non-trivial (Y; T )-periodic trajectories
for Y = 0 and it contains (0; 0)-periodic trajectories with non-constant projection onto
the (y; t) plane.
For some other physical examples of G!odel type manifolds where Theorems 1.6 and
1.7 hold, see, e.g., the introduction in [8].
2. Variational principle
Let M = M0 × R2 be a G!odel type space–time equipped with the Lorentzian metric
(1.1). Fix (Y; T ) ∈ R2 .
Our goal is to state a suitable variational principle which reduces the research of
(Y; T )-periodic trajectories in M to the study of critical points of a functional depending
only on the Riemannian component.
Taken I = [0; 1], let H 1 (I; M0 ) be the set of curves x : I → M0 such that for any
local chart (U; ’) of M0 , with U ∩ x(I ) = ∅, the curve ’ ◦ x belongs to the Sobolev
A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
613
space H 1 (x−1 (U ); Rn ); n = dim M0 . This implies that H 1 (I; M0 ) is equipped with a
structure of in4nite dimensional manifold modelled on the Hilbert space H 1 (I; Rn )
and for any x ∈ H 1 (I; M0 ) the tangent space to H 1 (I; M0 ) at x admits the following
identi4cation
Tx H 1 ≡ {& ∈ H 1 (I; T M0 ): &(s) ∈ Tx(s) M0 for all s ∈ I };
with T M0 tangent bundle of M0 . Let us remark that, by the Nash Imbedding Theorem
(cf. [15]), M0 can be smoothly isometrically imbedded in some Euclidean space RN ;
hence, H 1 (I; M0 ) is a submanifold of the Hilbert space H 1 (I; RN ) (for more details,
see [17]).
For simplicity, we will denote the Riemannian metric on M0 with ·; ·.
De4ne
1 (M0 ) = {x ∈ H 1 (I; M0 ): x(0) = x(1)}:
It is known that 1 (M0 ) is a Riemannian submanifold of H 1 (I; M0 ) whose tangent
space in x ∈ 1 (M0 ) is given by
Tx 1 (M0 ) = {& ∈ Tx H 1 : &(0) = &(1)}:
Moreover, 1 (M0 ) is complete if M0 is complete (i.e., if it is a complete submanifold
of the Euclidean space RN ).
Since we are interested in (Y; T )-periodic trajectories z = (x; y; t) such that y(0) = t(0)
= 0, let us de4ne
HY = {y ∈ H 1 (I; R): y(0) = 0; y(1) = Y };
HT = {t ∈ H 1 (I; R): t(0) = 0; t(1) = T }:
Assumed
Y∗ : s ∈ I → Ys ∈ R;
T∗ : s ∈ I → Ts ∈ R;
(2.1)
it is
HY = H01 + Y∗ ;
HT = H01 + T∗ ;
where
H01 = {u ∈ H 1 (I; R): u(0) = u(1) = 0}:
Then, HY ; HT are closed aKne submanifolds of the Hilbert space H 1 (I; R) and H01 is
the tangent space at every one of their points.
Classical variational tools imply that zU is a (Y; T )-periodic trajectory in M if and
only if zU is a critical point of the action functional
1
1
z;˙ z
˙ L ds
F(z) =
2 0
1 1
2
=
(ẋ; ẋ + A(x)ẏ 2 + 2B(x)ẏt˙ − C(x)t˙ ) ds;
2 0
z = (x; y; t), on the Hilbert manifold ZY; T ≡ 1 (M0 ) × HY × HT .
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A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
It is easy to prove that F is a C 1 functional on ZY; T and
1 1
˙
F (z)[.] =
ẋ; & ds +
(∇A(x); & ẏ 2 + 2A(x)ẏ/̇) ds
2 0
0
1
+
(∇B(x); &ẏt˙ + B(x)/̇t˙ + B(x)ẏ0̇) ds
1
0
1
−
2
1
0
2
(∇C(x); & t˙ + 2C(x)t˙0̇) ds
for any z = (x; y; t) ∈ ZY; T and . = (&; /; 0) ∈ Tz ZY; T ≡ Tx 1 (M0 ) × H01 × H01 .
Since the action functional F is unbounded both from above and from below, a new
functional on 1 (M0 ) is de4ned in such a way that it is bounded from below and its
critical points are related to those ones of F.
This variational approach has been introduced in [8] in order to study geodesics joining two given points in G!odel type manifolds. Anyway, for completeness, we outline
the main ideas of the proof.
First of all, let us de4ne
1y : 1 (M0 ) → HY
and
1t : 1 (M0 ) → HT
such that, if x ∈ 1 (M0 ), for all s ∈ I it is
s
B(x(2))
Yb(x) − Tc(x)
1y (x)(s) =
d2
L(x)
0 H(x(2))
Ya(x) + Tb(x) s C(x(2))
+
d2;
L(x)
0 H(x(2))
Yb(x) − Tc(x)
1t (x)(s) = −
L(x)
+
Ya(x) + Tb(x)
L(x)
s
A(x(2))
d2
H(x(2))
s
B(x(2))
d2:
H(x(2))
0
0
Clearly, 1y and 1t are two C 1 maps on 1 (M0 ).
Proposition 2.1. Let (M; ·; ·L ) be a G;odel type manifold such that (H2 ) holds. Then;
the following statements are equivalent:
(i) z ∈ ZY; T is a critical point of F;
(ii) z = (x; y; t) is such that x ∈ 1 (M0 ) is a critical point of the functional
1
J (x) =
2
0
1
ẋ; ẋ ds +
Y 2 a(x) + 2YTb(x) − T 2 c(x)
2L(x)
(2.2)
A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
615
and
y = 1y (x);
t = 1t (x);
where a; b; c and L are as in (1:4) and (1:5) and 1y ; 1t are as above.
Moreover; if (i) or (ii) holds; it is
F(x; 1y (x); 1t (x)) = J (x):
Proof. Since F is a C 1 functional on ZY; T , let us de4ne the partial derivatives of F
in z = (x; y; t) as follows:
Fx (z)[&] = F (z)[(&; 0; 0)]
for all & ∈ Tx 1 (M0 );
Fy (z)[/] = F (z)[(0; /; 0)]
for all / ∈ H01 ;
Ft (z)[0] = F (z)[(0; 0; 0)]
for all 0 ∈ H01 :
Clearly, all the critical points of F in ZY; T belong to
N = {z ∈ ZY; T : Fy (z) = Ft (z) = 0}:
(2.3)
Arguing as in the proof of [8, Lemma 2:3], it follows that N is a submanifold
of ZY; T , since it is the graph of the C 1 map 4 : 1 (M0 ) → HY × HT de4ned as
4(x) = (1y (x); 1t (x)) for every x ∈ 1 (M0 ).
Assume J is equal to the restriction of F to N , i.e.,
J (x) = F(x; 4(x))
for all x ∈ 1 (M0 ):
Then, it is easy to see that z ∈ ZY; T is a critical point of F, if and only if z = (x; y; t) ∈ N ,
i.e., (y; t) = 4(x), and J (x) = 0, as,
J (x)[&] = Fx (z)[&] = F (z)[(&; 0; 0)]
1
1
˙ ds + 1
=
ẋ; &
∇A(x); &ẏ 2 ds
2 0
0
1
1 1
2
˙
+
∇B(x); &ẏt ds −
∇C(x); &t˙ ds
2
0
0
(2.4)
for all & ∈ Tx 1 (M0 ).
Remark 2.2. Let M be a G!odel type space–time such that (H2 ) holds. If we look for
closed geodesics in M, i.e., geodesics in M which satisfy the boundary conditions
(1.3) with Y = T = 0, then, by Proposition 2.1 and the de4nitions of 1y ; 1t , a curve
zU = (x;
U y;
U tU) ∈ Z0; 0 is a critical point of F, if and only if yU = tU = 0 while xU is a critical
point of the functional
1 1
ẋ; ẋ ds on 1 (M0 ):
f(x) =
2 0
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A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
So, when (H2 ) holds, a curve zU = (x;
U y;
U tU ) in M is a closed geodesic, if and only if xU
is a closed geodesic in the Riemannian manifold M0 and y;
U tU are constant.
Now, let us recall the main de4nitions and results of the Ljusternik–Schnirelman
theory on manifolds (for more details, see, e.g., [18,22]).
Denition 2.3. Let X be a topological space. Given A ⊆ X; catX (A) is the Ljusternik–
Schnirelman category of A in X , that is the least number of closed and contractible
subsets of X covering A. Otherwise, it is cat X (A) = + ∞ if it is not possible to cover
A with a 4nite number of such sets.
We assume cat(X ) = catX (X ).
Denition 2.4. Let be a Riemannian manifold modelled on a Hilbert space. A
C 1 functional g : → R satis4es the Palais–Smale condition, brieMy (PS), if every
sequence (x n )n∈N in , such that
sup |g(x n )| ¡ + ∞
n∈N
and
lim g (x n ) = 0;
n→+ ∞
has a convergent subsequence (here, g (x n ) goes to 0 in the norm induced on the
cotangent bundle by the Riemannian metric on ).
Theorem 2.5 (Ljusternik–Schnirelman theorem). Let be a complete Riemannian
manifold and let g : → R be a C 1 functional which satis>es (PS). If k ∈ N; k ¿ 1;
let us de>ne
9k = {A ⊆ : cat (A) ¿ k};
(2.5)
ck = inf sup g(x):
(2.6)
A∈9k x∈A
If 9k = ∅ and ck ∈ R; then ck is a critical value of g.
If g is a functional on a manifold , for any c ∈ R we set
gc = {x ∈ : g(x) 6 c};
gc = {x ∈ : g(x) ¿ c}:
Remark 2.6. Let and g be as in Theorem 2.5. If g is bounded from below, then for
any c ∈ R it is
cat (gc ) ¡ + ∞:
(2.7)
As a consequence of the Ljusternik–Schnirelman theorem, we can prove the following propositions.
A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
617
Proposition 2.7. Let g be a C 1 functional on a smooth Riemannian manifold such
that cat() =+ ∞. Assume that (2:7) holds for all c ∈ R. If L ∈ R is such that gL = ∅;
then there exists kL ∈ N such that ckL ¿ L; where ckL is as in (2:6).
Proof. Let L ∈ R be such that gL = ∅. By the de4nition (2.6) it is enough to prove
that there exists kL ∈ N such that B ∩ gL = ∅ for all B ∈ 9kL , where 9kL is de4ned as
in (2.5).
In fact, otherwise, for any k ∈ N there exists Bk ∈ 9k such that Bk ∩ gL = ∅, then
Bk ⊂ g L ;
cat (Bk ) ¿ k ⇒ cat (g L ) ¿ k
in contradiction with (2.7).
Corollary 2.8. Let be a complete Riemannian manifold which has compact subsets
of arbitrarily high category. Let g : → R be a C 1 functional bounded from below
but not from above. Assume that g satis>es (PS). Then there exists an increasing
diverging sequence of critical levels of g on .
Proof. Let n ∈ N; n ¿ 1, be 4xed. Since g is not bounded from above, then the
“superlevel” gn is not empty; moreover, by Remark 2.6 the condition (2.7) holds
for all c ∈ R. By Proposition 2.7 and the topological assumptions on it follows that
there exist kn ∈ N and a compact set Kn ∈ 9kn such that
n 6 ckn 6 sup g(x) ¡ + ∞:
x∈Kn
So, Theorem 2.5 implies ckn is a critical level of g and
lim ckn = + ∞.
n→+ ∞
The following result, due to Fadell and Husseini (cf. [10]), allows to evaluate the
Ljusternik–Schnirelman category of 1 (X ), space of loops of an open subset of RN .
Anyway, standard arguments allow to apply such a result even when X = M0 is a
manifold.
Proposition 2.9. Let X be an open subset of RN which is connected; not contractible
in itself and such that its fundamental group 1 (X ) does not have the property that
it is an in>nite group with >nitely many conjugacy classes. Then cat(1 (X )) = + ∞.
Moreover; 1 (X ) possesses compact subsets of arbitrarily high category.
3. The compact case
Let M = M0 × R2 be a manifold of G!odel type equipped with the Lorentzian metric
(1.1) such that M0 is at least complete and the assumptions (H2 ); (H3 ) hold. Fix a
non-zero couple (Y; T ) ∈ R2 .
Since Proposition 2.1 applies, then (Y; T )-periodic trajectories in M can be searched
looking for critical points of the C 1 functional J : 1 (M0 ) → R de4ned in (2.2).
In order to apply Corollary 2.8 the following lemmas are useful.
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A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
Lemma 3.1. The functional J is bounded from below on 1 (M0 ).
Proof. The proof is an obvious consequence of the hypothesis (H3 ):
1
1 1
ẋ; ẋ ds − (k1 Y 2 + 2k2 |YT | + k3 T 2 ):
J (x) ¿
2
2 0
Remark 3.2. Taken k(Y; T ) = k1 Y 2 + 2k2 |YT | + k3 T 2 , the proof of Lemma 3.1 implies
1
ẋ; ẋ ds 6 2 J (x) + k(Y; T ):
(3.1)
0
Let us remark that trivial periodic trajectories in M correspond to constant critical
points of J in 1 (M0 ); hence, we need more information about J on constant curves.
De4ne hY; T : M0 → R such that
hY; T (x) = Y 2 A(x) + 2YTB(x) − T 2 C(x):
Clearly, hY; T is a C 1 map. Taken k ∈ M0 , by (1.4) and (1.5), on the corresponding
constant map in 1 (M0 ) there results
L(k) =
a(k)
= A(k);
L(k)
1
;
H(k)
b(k)
= B(k);
L(k)
c(k)
= C(k);
L(k)
(3.2)
hence,
J (k) = 21 hY; T (k):
(3.3)
Lemma 3.3. Taken k ∈ M0 ; the following statements are equivalent:
(i) k is a trivial critical point of J on 1 (M0 );
(ii) k is a critical point of hY; T on M0 .
Proof. By (3.2) it follows
1y (k) = Y∗ ;
1t (k) = T∗ ;
(3.4)
where 1y and 1t are as in Section 2 while Y∗ , T∗ are de4ned in (2.1).
So, (2.4) gives J (k) = 0 if and only if
1
∇hY; T (k); & ds = 0 for all & ∈ Tk 1 (M0 );
0
i.e., ∇hY; T (k) ≡ 0.
Remark 3.4. If (H2 ) holds, by Lemma 3.3 it follows that z : [0; 1] → M, z = (x; y; t),
is a trivial periodic trajectory in M if and only if x ≡ k is a critical point of hY; T
while y and t are straight lines of slope Y and T , respectively (see the de4nitions in
(3.4)). Let us point out that such a result does not need the assumption (H2 ) and can
also be proved just using the geodesic equations.
A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
619
Corollary 3.5. If x ∈ 1 (M0 ) is a critical point of J such that 2 J (x) ¿ k(Y; T ); then
x is not constant.
Proof. It is easy to see that
|A(x)| 6 k1 ;
|B(x)| 6 k2 ;
|C(x)| 6 k3
for all x ∈ M0
(trivial estimates if M0 is compact, otherwise coming from (3.2) and (H3 )) which
imply
sup |hY; T (x)| 6 k(Y; T ):
x∈M0
So, the result follows by (3.3) and Lemma 3.3.
Lemma 3.6. Let M0 be compact. Then the functional J satis>es (PS) in 1 (M0 ).
Proof. Let (x n )n∈N ⊂ 1 (M0 ) be such that
(J (x n ))n∈N is bounded
and
lim J (x n ) = 0:
(3.5)
n→∞
Clearly, by (3.1) it follows that
1
ẋn ; ẋn ds
is bounded;
0
n∈N
then, since M0 is compact, the sequence (x n )n∈N is bounded in H 1 (I; M0 ) and there
exists x ∈ H 1 (I; RN ) such that
x n * x weakly in H 1 (I; RN );
x n → x uniformly in I
(3.6)
(up to subsequences). Clearly, x ∈ 1 (M0 ). By [3, Lemma 2:1] there exists two bounded
sequences (&n )n∈N and (n )n∈N in H 1 (I; RN ); &n ∈ Tx n 1 (M0 ), such that
x n − x = &n + n
&n * 0 weakly
for all n ∈ N;
and
n → 0 strongly in H 1 (I; RN ):
(3.7)
Our aim is to verify that &n → 0 strongly in H 1 (I; RN ).
In fact, taken yn = 1y (x n ); tn = 1t (x n ) and zn = (x n ; yn ; tn ) ∈ N (see (2.3)), by (2.4)
and (3.5) it is
J (x n )[&n ] = F (zn )[(&n ; 0; 0)] = o(1);
i.e.,
o(1) =
0
+
1
ẋn ; &˙n ds +
0
1
1
2
0
1
∇A(x n ); &n ẏ2n ds
∇B(x n ); &n ẏ n t˙n ds −
1
2
0
1
2
∇C(x n ); &n t˙n ds:
(3.8)
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A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
Let us remark that, since the Palais–Smale sequence (x n )n∈N is bounded and (H3 )
holds, the de4nitions of 1y and 1t imply that
1
0
ẏ2n ds
and
n∈N
1
0
2
t˙n
ds
are bounded;
n∈N
so, (3.6) and (3.7) give
1
0
1
0
1
0
∇A(x n ); &n ẏ2n ds = o(1);
∇B(x n ); &n ẏ n t˙n ds = o(1);
2
∇C(x n ); &n t˙n ds = o(1):
Finally, since x n = x + &n + n , by (3.8) and
1
0
ẋ; &˙n ds = o(1);
0
1
˙n ; &˙n ds = o(1)
it follows that
0
1
&˙n ; &˙n ds = o(1):
Proof of Theorem 1.6. From the hypothesis (H1 ) it follows that Proposition 2.9
applies; moreover, since Lemmas 3.1 and 3.6 hold, by Corollary 2.8 there exists
an increasing diverging sequence of critical levels of J on 1 (M0 ). Clearly, by
Corollary 3.5, in4nitely many of the corresponding critical points, should not be
constant; hence, by Proposition 2.1, in4nitely many non-trivial (Y; T )-periodic
trajectories exist.
Remark 3.7. If M0 is compact but it does not satisfy (H1 ), the other hypotheses of
Theorem 1.6 imply the existence of a critical point of J for curves at any conjugacy
class C of the fundamental group of M0 . More precisely, let 1C be the subset of
1 (M0 ) containing the curves which lie in the conjugacy class C. It can be proved
that 1C is an open and closed subset of 1 (M0 ); thus, J attains a minimum point in
1C which is also a critical point in 1 (M0 ). If C is the trivial conjugacy class this
result is not interesting since J attains its minimum in the minimum points of hY; T , and
then the corresponding trajectories are trivial. But if (H1 ) does not hold, then at least
there is a non-trivial conjugacy class and, so, a non-trivial (Y; T )-periodic trajectory is
found.
A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
621
4. The complete case
Let M = M0 × R2 be a G!odel type manifold equipped with the Lorentzian metric
(1.1) such that M0 is complete and the assumptions (H2 ) and (H3 ) hold.
Fixed a non-zero couple (Y; T ) ∈ R2 , by Proposition 2.1 the (Y; T )-periodic trajectories in M are critical points of the functional J : 1 (M0 ) → R de4ned in (2.2).
Unluckly, since M0 is complete but in general unbounded, J could not satisfy the
Palais–Smale condition. In fact, a sequence (x n )n∈N in M0 may exist such that
U → + ∞ and hY; T (x n ) → 0 as n → + ∞ (for a certain xU ∈ M0 ) while in
d(x n ; x)
1 (M0 ) it is a sequence which satis4es the conditions (3.5) and has no converging
subsequences.
In order to overcome such a problem, the existence of a “convex at in>nity” map
allows to introduce a family of penalized functionals (F' )'¿0 in such a way that every
J' , associated to F' , satis4es (PS).
Thus, let U be such that the hypothesis (H5 ) holds. It can be proved that
∇U (x); ∇U (x)1=2 ¿ $ d(x; x0 ) − c1 ;
(4.1)
$ 2
d (x; x0 ) − c2 d(x; x0 ) − c3
(4.2)
2
for suitable positive constants c1 ; c2 ; c3 and any x ∈ M0 , where $ and x0 are as in
(H5 ) (cf. [5, Lemma 2:2]).
Fixed ' ¿ 0, let ' : R+ → R+ be a C 2 “cut-function” such that

1

0
if 0 6 s 6 ;



'
+
∞
n
' (s) =
 $n
1
1


s−
if s ¿ :

n!
'
'
U (x) ¿
n=3
It is easy to prove that there exist some positive constants a; b such that
' (s) ¿ $ ' (s) ¿ as
' (s) 6
' (s)
−b
for all s ∈ R+ ;
for all ' 6 ' ; s ∈ R+ :
(4.3)
(4.4)
For any x ∈ M0 , de4ne
U' (x) =
' (U (x)):
Then, the action functional F can be penalized by means of U' (x); more precisely,
de4ne
1
F' (z) = F(z) +
U' (x) ds; z = (x; y; t) ∈ ZY; T :
0
Clearly, F' is of class C 1 ; moreover, since the penalization term does not depend on
the variables (y; t), for all z = (x; y; t) ∈ ZY; T and /; 0 ∈ H01 it is
F' (z)[(0; /; 0)] = F (z)[(0; /; 0)];
F' (z)[(0; 0; 0)] = F (z)[(0; 0; 0)]:
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A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
Then, arguing as in Proposition 2.1, taken 1y and 1t as in Section 2, there results that
z' = (x' ; y' ; t' ) is a critical point of F' , if and only if x' is a critical point of
1
J' (x) = J (x) +
U' (x) ds in 1 (M0 );
(4.5)
0
while y' = 1y (x' ), t' = 1t (x' ). Moreover, J' (x' ) = F' (z' ) and
J' (x' )[&] = F' (z' )[(&; 0; 0)]
= F (z' )[(&; 0; 0)] +
0
1
' (U (x' ))∇U (x' ); & ds
for all & ∈ Tx' 1 (M0 ).
Let us point out that, since ' is positive, (3.1) implies
1
ẋ; ẋ ds 6 2J' (x) + k(Y; T ) for all x ∈ 1 (M0 );
0
(4.6)
(4.7)
hence, J' is bounded from below.
Lemma 4.1. For all ' ¿ 0 the functional J' satis>es (PS) in 1 (M0 ).
Proof. Fixed ' ¿ 0, let (x n )n∈N ⊂ 1 (M0 ) be such that
(J' (x n ))n∈N is bounded
and
lim J' (x n ) = 0:
n→∞
By (4.7) and (4.8) it follows that
1
ẋn ; ẋn ds
is bounded:
0
(4.8)
(4.9)
n∈N
If it is
sup{d(x n (s); x0 ): s ∈ I; n ∈ N} = + ∞
(taken x0 as in (H5 )), then
lim inf d(x n (s); x0 ) = + ∞
n→∞
s∈I
which implies, by (4.2), that
lim inf U (x n (s)) = + ∞:
n→∞
s∈I
Clearly, (4.3) and this last formula gives
1
U' (x n ) ds = + ∞:
lim
n→∞
0
(4.10)
A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
623
On the other hand, by (3.1) and (4.5) it follows
1
k(Y; T )
;
U' (x n ) ds 6 J' (x n ) +
2
0
then, by (4.8),
1
U' (x n ) ds
0
is bounded
n∈N
in contradiction with (4.10).
Thus, sup{d(x n (s); x0 ): s ∈ I; n ∈ N} ¡ + ∞; hence, by (4.9), (x n )n∈N is bounded in
H 1 (I; M0 ), so (3.6), (3.7) hold.
It is easy to verify that (4.6), (x n )n∈N bounded and &n * 0 weakly in H 1 (I; RN ) (see
(3.7)) imply
1
' (U (x n ))∇U (x n ); &n ds = o(1);
0
then, arguing as in the last part of the proof of Lemma 3.6, the conclusion
follows.
In order to avoid the penalization argument, the following proposition needs.
Proposition 4.2. Let M = M0 × R2 be a G;odel type manifold such that M0 is complete and the assumptions (H2 ) – (H6 ) hold. Taken any couple of positive constants
m; M with
k(Y; T )
¡m¡M
(4.11)
2
there exists '0 = '0 (m; M ) ¿ 0 such that for every ' ∈ ]0; '0 [ if x' ∈ 1 (M0 ) satis>es
J' (x' ) = 0;
m 6 J' (x' ) 6 M;
(4.12)
J (x' ) = 0:
(4.13)
then
J (x' ) = J' (x' );
Proof. Obviously, if there exists '0 ¿ 0 such that (4.12) implies
1
sup U (x' (s)) ¡ ;
'
s∈I
(4.14)
with ' 6 '0 , then ' (U (x' (s))) = ' (U (x' (s))) = 0 for all s ∈ I and (4.13) holds.
Thus, in order to prove (4.14), let us argue by contradiction and assume that for each
n ∈ N there exist 'n ¿ 0 and x n ∈ 1 (M0 ) such that 'n 0 and
J'n (x n ) = 0;
m 6 J'n (x n ) 6 M;
(4.15)
sup U (x n (s)) ¿
1
:
'n
(4.16)
s∈I
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A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
It is easy to prove that, since U is continuous, (4.16) implies
sup{d(x n (s); x0 ): s ∈ I; n ∈ N} = + ∞;
(4.17)
moreover, by (4.7) and (4.15) it follows that
1
ẋn ; ẋn ds
is bounded:
(4.18)
Clearly, (4.17) and (4.18) give
inf d(x n (s); x0 ) = + ∞;
lim
(4.19)
0
n∈N
n→+ ∞
s∈I
hence, by (4.2) it is
inf U (x n (s)) = + ∞:
lim
n→+ ∞
s∈I
Fixed n ∈ N, assume un (s) = U (x n (s)); yn = 1y (x n ); tn = 1t (x n ).
As J'n (x n ) = 0, by (4.6) and (2.4) classical arguments imply that x n is a C 2 map such
that ẋn (0) = ẋn (1) and
2
DsR ẋn = 12 ẏ2n ∇A(x n ) + ẏ n t˙n ∇B(x n ) − 12 t˙n ∇C(x n ) +
'n (U (x n ))∇U (x n );
where DsR ẋn is the covariant derivative of ẋn along x n induced by the Levi–Civita
connection of ·; · (e.g., cf. [14]).
Then un ∈ C 2 (I; R) and un (0) = un (1); u̇ n (0) = u̇ n (1), where for all s ∈ I it is
u̇ n (s) = ∇U (x n (s)); ẋn (s):
From the above equation and
u! n (s) = HRU (x n (s))[ẋn (s); ẋn (s)] + ∇U (x n (s)); DsR ẋn (s);
s ∈ I;
it follows
0=
u! n ds =
0
=
1
0
+
1
(HRU (x n )[ẋn ; ẋn ] + ∇U (x n ); DsR ẋn ) ds
HRU (x n )[ẋn ; ẋn ] ds +
1
0
+
0
1
0
1
1
2
0
1
∇U (x n ); ∇A(x n )ẏ2n ds
∇U (x n ); ∇B(x n )ẏ n t˙n ds −
1
2
1
0
'n (U (x n ))∇U (x n ); ∇U (x n ) ds:
2
∇U (x n ); ∇C(x n )t˙n ds
A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
625
Let us remark that (H4 ) and the de4nitions of 1y and 1t imply that
1
1
2
2
t˙n ds
ẏ n ds
and
are bounded;
0
0
n∈N
n∈N
then by (H6 ) and (4.19) if n → + ∞ we obtain
1
∇U (x n ); ∇A(x n ) ẏ2n ds ¿ o(1);
0
1
0
0
1
∇U (x n ); ∇B(x n ) ẏ n t˙n ds = o(1);
2
∇U (x n ); ∇C(x n )t˙n ds 6 o(1);
which gives
1
HRU (x n )[ẋn ; ẋn ] ds +
0¿
0
0
1
'n (U (x n ))∇U (x n ); ∇U (x n ) ds
+ o(1):
Furthermore, if n is large enough, for all s ∈ I by (4.1) and (4.19) it follows
∇U (x n (s)); ∇U (x n (s)) ¿ 2;
while (H5 ) implies
HRU (x n (s))[ẋn (s); ẋn (s)] ¿ $ẋn (s); ẋn (s):
Thus, if n is large enough, it is
1
1
ẋn ; ẋn ds + 2
0¿$
0
0
'n (U (x n )) ds
+ o(1):
(4.20)
On the other hand, by the de4nition of J' , (H3 ) and (4.15) imply
1
1 1
k(Y; T )
ẋn ; ẋn ds +
m 6 J'n (x n ) 6
+
'n (U (x n )) ds
2 0
2
0
(with k(Y; T ) de4ned in Remark 3.2), then (4.20) becomes
1
( 'n (U (x n )) − $ 'n (U (x n ))) ds + o(1)
0 ¿ $(2m − k(Y; T )) + 2
0
and, by (4.3), this last inequality is in contradiction with (4.11) as n goes to
in4nity.
Proof of Theorem 1.7. De4ned
1
ẋ; ẋ ds
f(x) =
0
for all c ∈ R the inequality (3.1) implies that J c ⊂ f2c+k(Y; T ) .
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A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
Since cat 1 (M0 ) (f2c+k(Y; T ) ) ¡+ ∞ (for more details, see [7, Lemma 4:1]), there results
cat1 (M0 ) (J c ) ¡ + ∞:
(4.21)
Fixed
m¿
k(Y; T )
;
2
(4.22)
by Proposition 2:9, (4.21) and J unbounded from above, Proposition 2.7 implies that
U
there exists kU ∈ N; kU = k(m),
such that
inf sup J (x) ¿ m;
(4.23)
A∈9kU x∈A
where 9kU is de4ned in (2.5).
On the other hand, for all ' ∈ ]0; 1] by (4.4) and (4.5) it is
J (x) 6 J' (x) 6 J1 (x)
for all x ∈ 1 (M0 );
hence, if we assume
c'; k = inf sup J' (x)
A∈9k x∈A
for any ' ¿ 0; k ∈ N, (4.23) implies that
m 6 c'; kU 6 c1; kU:
(4.24)
Furthermore, by Proposition 2.9 there exists a compact set KU in 1 (M0 ) such that
U so (4.24) implies
U ¿ k,
cat1 (M0 ) (K)
m 6 c'; kU 6 M
where M = max J1 (x); ' ∈ ]0; 1]:
x∈KU
(4.25)
By Lemma 4.1 and Theorem 2.5 it follows that c'; kU is a critical level of J' ; moreover,
Proposition 4.2 and (4.22), (4.25) imply that c'; kU is a critical level of J , too, if ' 6 '0
for a certain '0 ∈ ]0; 1].
Let us point out that any corresponding critical point xkU of J is not trivial as a consequence of (4.22), (4.25) and Corollary 3.5.
Thus, constructing two sequences (mi )i∈N and (Mi )i∈N , where
m0 = m;
M0 = M;
mi ¡ Mi ¡ mi+1
for all i ∈ N;
such that at any couple mi ; Mi the previous arguments apply, we can 4nd a sequence of
critical points of J whose critical levels diverge positively; whence, by Proposition 2.1,
in4nitely many non-trivial (Y; T )-periodic trajectories exist on M.
Remark 4.3. Let M0 be complete and such that (H2 )–(H6 ) are satis4ed. Arguing
as in the compact case (see Remark 3.7), if (H1 ) does not hold but M0 is complete
and not simply connected, then at least one non-trivial (Y; T )-periodic trajectory exists
on M.
A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
627
5. Examples
5.1. Killing vector >elds with non-constant causal character
Let M0 be a compact Riemannian manifold (or a manifold satisfying the hypotheses
(H1 ) and (H5 ) of Theorem 1.7) and choose any diWerentiable mapping : M0 → R
whose range contains the interval [0; ], i.e., [0; ] ⊆ (M0 ). Put
A(x) ≡ C(x) ≡ sin (x);
B(x) ≡ ’((x));
where ’ : R → R is a function. Notice that the Killing vector 4elds @y ; @t do not have a
constant causal character on all M. If m := inf (’) ¿ 0 then, clearly, there exists ¿ 0
such that L(x) ¿ for all x ∈ 1 (M0 ). So, by Remark 1.9, Theorem 1.6 is applicable
(recall also that if M0 is just complete, the hypothesis (H 4) of Theorem 1.7 will also
hold).
5.2. Warped and static cases
Consider a G!odel type manifold such that all the coeKcients A; B; C are equal up to
a multiplicative constant. That is, there exist a C 1 real function A on M0 and some
real constants B1 ; B2 ; B3 , such that the coeKcients A; B; C satisfy
A(x) = B1 A(x);
B(x) = B2 A(x);
C(x) = B3 A(x)
and B22 +B1 B3 ¿ 0; A(x) = 0 for all x ∈ M0 (A will be assumed positive, without loss of
generality). This manifold is a warped
space–time (see, for example, [16, Chapter 7]),
with base M0 , warping function A and 4ber (R2 ; gF ≡ B1 dy2 + 2B2 dy dt − B3 dt 2 ).
Recall that the 4ber is isometric to L2 (in fact, no loss of generality would occur if
B1 = B3 = 1; B2 = 0 are assumed). The projection of any geodesic on the (y; t)-plane
is a reparametrization of a straight line (it can be checked directly from the expressions for 1y and 1t , see below). For any (Y; T )-periodic trajectory, the sign of
gF ((Y; T ); (Y; T )) = Y 2 B1 + 2YTB2 − T 2 B3 ≡ K(Y; T ) determines if this straight line in
the 4ber is timelike, spacelike or lightlike.
Recall that, now, it is
2
1
1
1
L(x) = 2
ds ¿ 0
B2 + B1 B3
0 A(x)
for all x ∈ 1 (M0 ). So, the condition (H2 ) always holds. Moreover, if ' ¡ A ¡ M , for
some '; M ¿ 0, then (H3 ) and (H4 ) also hold. In particular, if M0 is compact then
in4nitely many (Y; T )-periodic trajectories exist for any (Y; T ) ∈ R2 \ {(0; 0)}. Notice,
also, that an existence result for timelike geodesic is obtained whenever YT → −∞.
In this case, the variational setting is similar to that one which is found looking for
periodic trajectories in a static Lorentzian manifold. In fact, by Proposition 2.1, the
functional J on 1 (M0 ) becomes
−1
1
1 1
1
K(Y; T )
J (x) =
ds
ẋ; ẋ ds +
:
(5.1)
2 0
2
0 A(x)
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A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
So, according to the choice of the periods (Y; T ), diWerent situations occur: if K(Y; T )
= 0, then J is just the energy functional of the closed geodesics in the Riemannian
manifold M0 ; if K(Y; T ) ¡ 0, then J is essentially equal to the functional introduced
in the study of geodesics in static Lorentzian manifolds (cf. [4]); if K(Y; T ) ¿ 0; J is
positive hence the study of its critical points is easier.
From a more general viewpoint, consider any complete semi-Riemannian manifold
(F; gF ), a positive function A on M0 and the warped product (M0 × F; g) where, under
natural identi4cations, g = ·; ·R + AgF . Let 0 : M0 × F → M0 ; F : M0 × F → F be
the natural projections. It is well-known that, for any geodesic C of the warped product,
the projection F ◦ C is a pregeodesic (i.e., a geodesic up to a reparametrization) of
(F; gF ). Thus, choose a geodesic Ĉ ≡ Ĉ(t) of (F; gF ). We will say that the geodesic
C of (M0 × F; g) is T -periodic along Ĉ if F ◦ C(s) = Ĉ(t(s)) for a reparametrization
t = t(s) such that
t(0) = 0;
t(1) = T;
t˙(0) = t˙(1):
If Ĉ is lightlike then 0 ◦ C is a geodesic in M0 and the reparametrization t(s) must
be any aKne function. As a consequence, 4xed a lightlike Ĉ and choosing 0 ◦ C as
a closed geodesic, then a T -periodic trajectory along Ĉ is obtained for every T ∈ R
(giving rounds to 0 ◦ C, in4nitely many such trajectories are also found, which are
geometrically distinct in the space–time, if T = 0). When Ĉ is timelike (respectively,
spacelike) then the problem is equivalent to study the functional (5.1) for K(Y; T ) ¡ 0
(respectively, K(Y; T ) ¿ 0). In particular, results analogous to those of Theorems 1.6
and 1.7 (with A ≡ C ≡ 0; B ≡ A) can be stated in this case.
Static space–times can be regarded as warped space–times with (F; gF ) = (R; −dt 2 ) and
the standard choice of Ĉ is Ĉ(t) = t. In particular, we obtain the existence of in4nitely
many T -periodic trajectories either when M0 is compact or when it is not contractible,
with 0 ¡ inf A ¡ sup A ¡ + ∞ and a function U as in Theorem 1.7 exists.
Nevertheless, this result can be strenghtened if T -periodic trajectories in a static space–
time (M0 × R; g = ·; ·R − A dt 2 ) are regarded as (0; T )-periodic trajectories in a G!odel
type space–time M = M0 × R2 with A ≡ 1; B ≡ 0; C ≡ A. In this case the conditions
(H3 ) and (H4 ) are automatically satis4ed if A = A(x) is bounded and far from zero. So,
when M0 is not compact, the only relevant hypotheses of Theorem 1.7 are (H1 ); (H5 )
and the last one of (H6 ). More precisely:
Corollary 5.1. Let (M0 × R; g); g = ·; ·R − A dt 2 ; be a static space–time such that
0 ¡ inf A ¡ sup A ¡ + ∞ and
(H1 ) (M0 ; ·; ·R ) is complete; not contractible in itself and its fundamental group
1 (M0 ) is >nite or it has in>nitely many conjugacy classes;
(H5 ) there exist x0 ∈ M0 ; U ∈ C 2 (M0 ; R+ ) and some positive constants R; #; $; such
that
x ∈ M0 ;
(H6 )
d(x; x0 ) ¿ R ⇒ HRU (x)[&; &] ¿ $&; &R
taken x0 and U as in (H5 ); there results
lim sup ∇U (x); ∇A(x)R 6 0:
d(x; x0 )→+ ∞
for all & ∈ Tx M0 ;
A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
629
Then; for any T ∈ R there exist in>nitely many non-trivial spacelike distinct T -periodic
trajectories in M0 × R whose energies diverge positively.
Clearly, the hypotheses (H5 ); (H6 ) can be dropped in the compact case.
5.3. Classical G;odel space–time (CGS)
Now, our goal is to study periodic trajectories in CGS. We must emphasize that previous results cannot be applied to this space–time, especially because their coeKcients
A; B; C do violate some of the hypotheses of our theorems and, in particular, among
them, the essential condition (H2 ). The following facts were shown in [8], where G!odel
type space–times were studied in order to determine their geodesic connectedness:
(a) even when violations of the hypotheses of variational theorems less severe than
those in CGS occur, variational methods may be non-applicable and geodesically disconnected space–time may appear (in fact, an explicit counterexample was given which
satis4ed (H2 ) but not (H3 ); none of these conditions are satis4ed by CGS); (b) nevertheless, CGS is geodesically connected, because of the symmetries and special characteristics of the metric.
Our study of periodic trajectories in CGS is motivated not only by the importance of
this classical space–time but also because it provides an example of the role of some
of our hypotheses.
We will use the study of geodesics in CGS carried out in [8], and summarize the
consequences for (Y; T )-periodic trajectories.
First of all, recall that CGS is R4 equipped with a G!odel type metric where M0 = R2
is the standard Euclidean space and
·; ·L = d x12 + d x22 − 12 e2Bx1 dy2 − 2eBx1 dy dt − dt 2 ;
√
with x = (x1 ; x2 ) ∈ R2 and B ¿ 0 given constant (! = B= 2 is the magnitude of the vorticity of the Mow). This metric is a semi-Riemannian product, and we will drop
√ the irrelevant factor (R; d x22 ). After a change of coordinates obtained by putting x = 2=Be−Bx1 ,
CGS can be written as (M; ·; ·L ) with
(M0 ; ·; ·R ) = (R+ ; (d x=(Bx)2 );
·; ·L =
√
2
1
1
2
2
d
x
−
dy
−
2
dy dt − dt 2 ;
(Bx)2
(Bx)2
Bx
(5.2)
where (x; y; t) ∈ R+ × R2 .
It is easy to see that none of the conditions (H1 )–(H4 ) of Theorem 1.7 is satis4ed by
CGS. Geodesics of CGS can be divided into the following three classes:
1. Trivial trajectories with Y = 0. Recall that if Y = 0 then K(Y; T ) is independent of
x; thus, a trivial trajectory can be found (recall that this is true even if (H2 ) does
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A.M. Candela et al. / Nonlinear Analysis 51 (2002) 607 – 631
not hold, see Remark 1.5). These (0; T )-periodic trajectories are obtained by putting
(x(s); y(s)) ≡ (x0 ; y0 ) (constant), t(s) ≡ sT .
2. Geodesics whose projection on the (x; y) half-plane R+ × R is a straight line or a
piece of circumference (this piece of circumference can be extended to a circumference
of the plane R2 ⊇ R+ × R). If this projection is a straight line parallel to the y-axis, a
trivial (Y; T )-periodic trajectory with Y = 0 is obtained. Otherwise, these geodesics are
not (Y; T )-periodic trajectories, even though they are complete.
3. Geodesics whose projection on the (x; y) half-plane R+ × R is a complete circumference. All these geodesics are (non-trivial) (Y; T )-periodic trajectories with Y = 0,
and they exist for all the values of T ∈ R. In particular, there exist closed geodesics
(i.e., (0; 0)-periodic trajectories) such that their projections on M0 are not closed
geodesics (this happens because of the violation of the hypothesis (H2 ), see
Remark 2.2).
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