PORTUGALIAE MATHEMATICA
Vol. 51 Fasc. 2 – 1994
A HADAMARD TYPE THEOREM
FOR THE STATIC SPACE-TIME
Walter Frattarolo
Abstract: Necessary and sufficient conditions on the static space-time are determined, in order that the exponential function expp be a diffeomorphism.
It is well known that the Hopf–Rinow theorem states that a connected and
complete Riemannian manifold is geodesically connected.
We can express this concept saying that, if M0 is such a manifold, then the
exponential map
expp : Tp M0 → M0
is onto, for every p ∈ M0 .
Hadamard theorem goes beyond this result and shows that, if M0 is simply
connected and complete and if its sectional curvature is k ≤ 0, then the map exp p
is a diffeomorphism, for every p ∈ M0 ; in particular:
i) There are not couples of conjugate points on M0 ;
ii) M0 is diffeomorphic to IRn (n = dim M0 );
iii) For any couple of points p, q ∈ M , there exists a unique geodesic
γ : IR → M0
such that
³
´
γ(0) = p; γ(1) = q .
The property (iii) above implies not only the existence of γ, that is to claim
expp is onto, but also the uniqueness of it, that is: expp is one to one.
No result like Hadamard theorem is known about semi-Riemannian manifolds;
on the contrary, there are counter-examples to the Hopf–Rinow theorem in the
case of static Lorentz manifolds (e.g.: see the anti-De Sitter space, on [9], [13]).
Received : February 7, 1992; Revised : May 22, 1992.
258
W. FRATTAROLO
However sufficient conditions which guarantee the validity of the Hopf–Rinow
theorem for static space-times, have been established in [7].
The Lorentz metric g = g(P ) considered there applies to a manifold M of the
following form:
M = M0 × IR ,
where M0 is furnished with a Riemannian metric (index = 0) denoted by: h · , · iM0
and g(P ) is defined by:
h
i
g(P ) (ζ, τ ), (ζ, τ ) = hζ, ζi − β(P0 ) τ 2
for every P = (P0 , t0 ) ∈ M and (ζ, τ ) ∈ Tp M ; here β is a positive real function
on M0 .
In the following g(P ) will be denoted by h · , · iM , if the context is clear.
In this work we examine necessary and sufficient conditions on the metric g(P )
for the function expp to be a diffeomorphism, for any P ∈ M . Such conditions, on
the contrary of what might be expected, are not a straightforward generalization
of the Hadamard ones. Before showing our results, we recall a result of [7]. Some
notation now.
Let P = (x0 , t0 ), Q = (x1 , t1 ) be two events in M . We shall consider loop
spaces Ω1 on M0 and M , defined in the following way:
n
Ω1 = Ω1 (M0 , x0 , x1 ) = x : [0, 1] → M0 : x absolutely continuous and such that
Z
1
o
hẋ, ẋiM0 ds < +∞, x(0) = x0 , x(1) = x1 .
0
Ω1 is a Riemannian manifold, modelled on Sobolev spaces, of curves in M0 .
The geodesic on M , joining P and Q, are the critical points of the action
functional f , defined by:
f (γ) =
Z
1
hγ̇(s), γ̇(s)iM ds .
0
In [7] the following variational principle has been proved:
Let M = M0 × IR be a static space-time with the metric h · , · iM .
Let P = (x0 , t0 ), Q = (x1 , t1 ) two points in M .
A curve γ(s) = (x(s), t(s)) is a critical point of f (i.e.: a geodesic) if and only
if:
·) x = x(s) is a critical point of the functional J on Ω1 , defined by:
J(x) =
Z
1
0
∆2
hẋ, ẋiM0 ds − R 1 ds ,
0 β(x)
A HADAMARD TYPE THEOREM FOR THE STATIC SPACE-TIME
259
where: ∆ = t1 − t0 ;
··) t = t(s) is the function such that:
t(0) = 0
and
ṫ(s) = ∆
³Z
1
(1/β(x)) ds
0
´−1
(1/β(x(s))) = Φ(x) .
Moreover if γ(s) = (x(s), t(s)) is a geodesic, then we have:
f (γ) = J(x) .
This variational principle allows to use the Riemannian techniques for problems over the manifold M and allows to apply the Lusternik and Schnirelman
critical point theory too.
We want also to recall a result obtained in [7] which is of interest for what
follows.
Using the same symbols as above, if M0 is connected and complete and if
β ∈ C 3 (M0 ; IR+ ) is a function limited from above and from below by positive
constants, then the Lorentz manifold (M, g) is geodesically connected.
In the sequel we shall write simply:
J(x) = Φ1 (x) − ∆2 Φ2 (x) ,
where:
Φ1 (x) =
Z
1
hẋ, ẋiM0 ds ,
0
Φ2 (x) = Z
1
1
0
ds
β(x)
.
Our main result is the following.
Theorem (1). Let M , M0 , IR, g, be defined as above. Suppose that:
i) M0 is simple connected and complete;
ii) β ∈ C 3 (M0 ; IR+ ) is a function limited from above and from below by
positive constants;
iii) the sectional curvature on M0 is k ≤ 0.
Under such assumptions, if the function expp : Tp M → M is a diffeomorβ
phism for any point P ∈ M , then the Riemannian Hessian HR
(P0 ) is a negative
semidefinite quadratic form, for any point P0 ∈ M0 .
β
(P0 ) is a negative definite quadratic
Viceversa, if the Riemannian Hessian HR
form for any point P0 ∈ M0 , then the function expp is a diffeomorphism for any
point P ∈ M .
260
W. FRATTAROLO
Now we get to prove Theorem (1) and some complementary results.
Proposition (2). Let : β ∈ C 2 (M0 ; IR+ ); P0 ∈ M0 ; R be the curvature
tensor on M . Let τ1 be the unitary vector field along IR, tangent to IR. The
following results are valid:
³
´2
p
1 β
i) hRv,τ1 v, τ1 i(P0 ,t0 ) = HR
(P0 )[v, v]− hv, grad βiP0 , ∀ t0 ∈ IR, ∀ v ∈ TP0 M0 .
2
β
ii) HR
(P0 ) is a negative semidefinite form iff:
³
hRv,τ1 v, τ1 i(P0 ,t0 ) ≤ − hgrad
∀ t0 ∈ R, ∀ v ∈ TPO M0 .
p
β, viP0
´2
,
Proof: (ii) is a straightforward consequence of (i). In order to prove (i), we
recall that:
√
−1
β
HR (P0 )[v, v] · hτ1 , τ1 iM
hRv,τ1 , v, τ1 i(P0 ,t0 ) = p
β(P0 )
¶
µ
√
−1
β
p
H (P0 )[v, v] · (−β(P0 ))
=
β(P0 ) R
√
q
β
= β(P0 ) HR (P0 )[v, v] , ∀ v ∈ TP0 M0 .
Now we observe that:
√
E
D
p
β
(2) HR (P0 )[v, v] = Dv (grad β), v
=
1
1
Dv √ grad β , v
2
β
µ
¿
1
=
2
*µ¿
v, grad p
µ
= −
1
4
¶
µ
1
4
¶
= −
¶
À
1
β(P )
À
1
β(P0 )
p
β(P0 )
³
β(P0 )
p
β(P0 )
³
1
(i) follows from (1) and (2).
Lemma (3). Assume:
P0
=
P0
M0
¶
grad β + p
hv, grad βiP0
hv, grad βiP0
´2
´2
1
Dv (grad β), v
β(P )
À
P0
1
+ p
hDv (grad β), viP0
2 β(P0 )
1
+ p
H β (P0 )[v, v], ,
2 β(P0 ) R
∀ v ∈ T P0 M 0 .
A HADAMARD TYPE THEOREM FOR THE STATIC SPACE-TIME
261
i) M0 is simply connected and complete;
ii) β ∈ C 3 (M0 ; IR+ ) is a function limited from above and from below by
positive constants;
iii) the sectional curvature on M is K ≤ 0.
β
If the Riemannian Hessian HR
(P0 ) is a negative definite quadratic form for
any point P0 ∈ M0 , then the quadratic form J 00 (x) is positive definite for any
curve x = x(s) (s ∈ I) which is a critical point of the functional J and for any
couple of extreme points x(0), x(1) ∈ M0 .
Proof: Let x be a critical point of J. We shall show:
1) J 00 (x) is a positive definite form if Φ002 (x) is negative definite;
2) Φ002 (x) is negative definite if the integral functional:
I(x)[v, v] =
Z
1
0
β
HR
(x)[v, v] ds ,
whose domain is the set of vector fields with compact support, tangent to M 0
along x, is a negative definite form.
If v = v(s) is a vector field having compact support and tangent to M0 along
x, we have:
·)
Φ1 (x) =
Φ001 (x)[v, v]
··)
=2
Z
Z
1
0
hẋ, ẋix(s) ds ;
1
hDs v, Ds vix(s) ds − 2
0
Z
1
0
hRẋ,v ẋ, vix(s) ds .
Besides that, we have:
1
·)
Φ2 (x) = R 1
··)
Φ02 (x)[v] = ³R
1
ds
0 β(x(s))
1
ds
0 β(x(s))
· · ·)
+ ³R
1
,
Φ2 ”(x)[v, v] = ³R
1
1
ds
0 β(x(s))
´2
Z
0
2
´2
Z
1
β 2 (x(s))
0
´3
ds
0 β(x(s))
´2
³
1
hgrad β, vix(s)
(−2)
β 3 (x(s))
hgrad β, vix(s)
µZ
1
ds ,
hgrad β, vix(s)
β 2 (x(s))
0
ds + ³R
1
1
ds
0 β(x(s))
´2
Z
ds
1
0
¶2
β
HR
(x)[v, v]
ds .
2
β (x(s))
262
W. FRATTAROLO
Now (1) comes straight, because:
J 00 (x) = Φ001 (x) − ∆Φ002 (x) .
Also (2) is soon found, as a consequence of the following
2
³R
1
ds
0 β(x(s))
= ³R
1
´3
µZ
1
hgrad β, vix(s)
β 2 (x(s))
0
2
ds
0 β(x(s))
´2
(
1
R1
ds
0 β(x(s))
ds
µZ
¶2
2
− ³R
1
ds
0 β(x(s))
´2
≤ ³R
1
2
ds
0 β(x(s))
´2
1
0
0
β 1/2 (x(s)) β 3/2 (x(s))
0
ds
β(x(s))
1
hgrad β, vix(s)
1
−
(µZ
Z
Z
1
0
³
Z
hgrad β, vix(s)
β 3 (x(s))
−
Z
1
0
1
0
´2
³
³
hgrad β, vix(s)
¶2
³
β 3 (x(s))
ds =
+
hgrad β, vix(s)
β 3 (x(s))
¶
´2
ds · R 1
1
ds
0 β(x(s))
hgrad β, vix(s)
β 3 (x(s))
´2
´2
ds
)
≤
+
ds
)
=0.
Lemma (4). Assume:
i) M0 is simply connected and complete;
ii) β ∈ C 3 (M0 ; IR+ ) is a function limited from above and from below by
positive constants;
iii) the sectional curvature on M0 is k ≤ 0.
If the quadratic form J 00 (x) is positive definite, for any curve x = x(s) (s ∈ I)
which is a critical point of the functional J, and for any couple of extreme points
β
x(0), x(1) ∈ M0 , then the Riemannian Hessian HR
(P0 ) is a negative semidefinite
quadratic form, for any point P0 ∈ M0 .
Proof: Let P0 ∈ M0 . By [7], there exists a geodesic γ(s) = (x(s), t(s)),
s ∈ I, contained in the manifold M and starting from P0 . Such geodesic depends
on Cauchy data for a system of second order differential equations, then it depends on
the point
P , on the initial instant t(0) = t0 and on the initial velocity
¡
¢ 0
γ̇(0) = ẋ(0), ṫ(0) . Now, as we are dealing with a static metric, it results (see
A HADAMARD TYPE THEOREM FOR THE STATIC SPACE-TIME
263
introduction):
t(s) = t(0) +
Rs
dρ
0 β(x(ρ))
dρ
β(x(ρ))
(
∆,
ṫ(s) = R 1
1
dρ
0 β(x)
·
s∈I
∆ = t(1) − t(0) ,
∆
,
β(x(s))
then ṫ(s) depends on x(s) and on ∆; therefore γ(s) depends on the vector
µ1 = ẋ(0), on ∆ and on the point (P0 , t0 ) ∈ M .
Fixing t(0) = t0 and ∆, we fix the temporal extrema of the geodesic arc γ(s),
s ∈ I, whereas fixing P and µ1 , we fix its spatial extrema; the vector µ(0) = µ0 ,
such that:
µ1 = ẋ(0) = kẋ(0)kTP0 M0 µ0
determines the direction of γ.
Stated that, let µ1 = ẋ(0) and ∆ be fixed and γ = γµ1 ;∆ be the corresponding
geodesic, with extreme points:
P = γ(0) = (x(0), t(0)) ,
x(0) = P0 ,
Q = γ(1) = ((x(1), t(1)) .
The curve x = x(s), s ∈ I is a critical point of J. Let ν0 ∈ TP0 M0 be a unitary
vector and let:
n
ν = ν(s) | ∀ s ∈ I : ν(s) ∈ Tx(s) M0 ; kν(s)kTx(s) M0 = 1
o
be the unitary vector field, obtained by parallel translation of ν0 along x. We
consider now a sequence of functions:
(1)
s : I → IR
sk (σ) =
and set:
k ∈ IN
1
1
σ+
k(k + 1)
k+1
xk (σ) = x(sk (σ)) ,
tk (σ) = t(sk (σ)) ;
from (1) it follows:
ẋk (σ) =
1
ẋ(s) |s=sk (σ) ,
k(k + 1)
ṫk (σ) =
1
ṫ(s) |s=sk (σ) ;
k(k + 1)
264
W. FRATTAROLO
besides that, we have:
∆k =
Z
=
Z
=
1
Z
ṫk (σ) dσ =
0
1/k
ṫ(s) ds =
1/k
1/k+1
Z 1/k
µ
ṫ(s)
k(k + 1) ds
k(k + 1)
¶
c
ds
β(x)
1/k+1
1/k+1
R 1/k
1
1/k+1 β(x) ds
R1 1
0 β(x) ds
∆;
here, we have used: β(x) ṫ = c (see the introduction).
The curve γk = (xk , tk ) are monotone linear reparametrizations of pieces of
γ = (x, t), therefore they are all geodesics and the sequence {xk } is made of
critical points of J.
Now we use an arbitrarily fixed function ϕ ∈ C0∞ (I, IR+ ) and build up the
following vector fields with compact support:

 vk (s) = vk (x(s)) |s=s (σ) = ϕ(σk (s)) ν(s),
k
1
k+1
≤ s ≤ k1 ,
1
s ∈ I\[ k+1
, k1 ] ;
 vk (s) = 0,
here, σk is the inverse function of sk .
We can also write: vk (σ) = ϕ(σ) ν(sk (σ)); then observe that:
Z
1
0
hDσ vk , Dσ vk ixk (σ) dσ =
=
Z
1
hDẋk vk , Dẋk vk ixk (σ) dσ =
0
Z 1D
0
=
Z 1D
0
³
´
Z 1µ
0
´E
dϕ
dσ
¶2
E
xk (σ)
1
dσ =
k(k + 1)
dσ
Z
1/k
1/k+1
µ
dϕ
ds
¶2
ds ,
indeed:
hẋk , ϕixk (σ) =
¿
=
¿
x∗k
µ
xk (σ)
dσ
ϕ̇(σ) ν(sk (σ)) + ϕ(σ) Dẋk ν(sk (σ)), ϕ̇(σ) ν(sk (σ))
+ ϕ(σ) Dẋk ν(sk (σ))
=
³
Dẋk ϕ(σ) ν(sk (σ) , Dẋk ϕ(σ) ν(sk (σ)
d
,ϕ
dσ
¶
d
, ϕ ◦ xk
dσ
À
xk (σ)
À
xk (σ)
=
dϕ(σ)
d
ϕ(xk (σ)) =
.
dσ
dσ
265
A HADAMARD TYPE THEOREM FOR THE STATIC SPACE-TIME
Therefore, we have (see Lemma (3)):
J 00 (xk )[vk , vk ] = Φ001 (xk )[vk , vk ] − ∆2k Φ002 (xk )[vk , vk ] =
(3)
=2
Z
1
0
∆2k
−
Z
hDσ vk , Dσ vk ixk (σ) dσ − 2
(
2
³R
´3
1 dσ
0 β(xk )
µZ
1
hRẋk ,vk ẋk , vk ixk (σ) dσ +
0
hgrad β, vk ixk (σ)
1
2
− ³R
1
´2
dσ
0 β(xk )
Z
1
0
³
whence:
2
Z 1µ
0
dϕ
dσ
¶2
dσ − 2
Z
> 2 R1
∆k
dσ
0 β(xk )
¶2 (
´2
Z
1
0
dσ +
1
H β (xk )[vk , vk ] dσ
2
β (xk ) R
)
>0,
1
0
hRẋk ,vk , ẋk , vk ixk (σ) dσ +
−
µ
´2
β 3 (xk )
1
+ ³R
1
+
hgrad β, vk ixk (σ)
dσ
0 β(xk )
(4)
dσ
β 2 (xk )
0
¶2
1
R1
dσ
0 β(xk )
µ
µZ
∆k
R1
1
dσ
0 β(xk )
¶2 Z
1
0
1
β 2 (x
hgrad β, vk ixk (σ)
β 2 (xk )
0
−
Z
1
0
k)
dσ
³
β
HR
(xk )[vk , vk ] dσ >
¶2
+
hgrad β, vk ixk (σ)
β 3 (xk )
´2
dσ
)
and using (2):
(5)
2
Z 1µ
0
dϕ
dσ
−
1
>2
k(k + 1)
µ
µ
¶2
dσ − 2
1
k(k + 1)
¶2 µ
1
0
¶2 µ
∆
R1
Z
ds
0 β(x)
hRẋk ,vk , ẋk , vk ixk (σ) dσ +
∆
R1
ds
0 β(x)
¶2 (
¶2
1
β
HR
(xk )[ν, ν]
inf
2
σ∈I β (xk )
k(k + 1)
R 1/k
ds
1/k+1 β(x)
µ
µZ
1/k
¶Z
hgrad β, vk ix(s)
1/k+1
− k(k + 1)
β 2 (x)
Z
1/k
1/k+1
1
ϕ2 (σ) dσ >
0
ds
¶2
+
hgrad β, vk ix(s)
β 3 (x)
)
ds .
266
W. FRATTAROLO
Afterwards, we are going to apply the mean theorem for continuous functions to
the second part of inequality (5); so we obtain:
(6)
2
Z 1µ
0
dφ
dσ
−
>2
µ
¶2
µ
∆
R
k(k+1) 01
dσ − 2
Z
1
0
∆
R
k(k + 1) 01
ds
β(x)
¶2 (
hRẋk ,vk ẋk , vk ixk (σ) dσ +
ds
β(x)
(k)
¶2
β(x(s1 ))
inf
1
≤s≤ k1
k+1
³
µ
1
H β (x)[ν, ν]
2
β (x) R
hgrad β, vk ix(s(k) )
2
(k)
β 4 (x(s2 ))
´2
−
³
¶Z
1
ϕ2 (σ) dσ >
0
hgrad β, vk ix(s(k) )
3
(k)
β 3 (x(s3 ))
´2
)
;
1
, k1 [ are suitable points.
here, the si ∈ ] k+1
At last observe that, by the completeness of M , the geodesic γ = γ(s) can be
extended from s ∈ I up to s ∈ IR; after making that, let’s divide the inequality
(6) by ∆2 and go to the limit for (∆ → +∞), then let’s multiplicate the resulting
inequality by (k(k + 1))2 and go again to the limit for (k → +∞). We shall
obtain the thesis of the lemma.
Proposition (5). We suppose that
1) γ = (x, t) is a geodesic contained in M ;
2) f (γ) =
R1
0
hγ̇, γ̇iM ds;
3) Φ(x) = t is the function obtained in [7] ( see the introduction);
let us denote
4) v a vector field having compact support and tangent to M0 along x;
5) (w, τ ) a vector field having compact support and tangent to M along γ.
Then:
h
i
J 00 (x)[v, w] = f 00 (γ) (v, Φ0 (x)[v]), (w, τ ) .
Proof: On smooth curves y = y(s) contained in M0 , it is identically (see
[7]):
(1)
ft0 (y, t) |t=Φ(y) = 0 ;
therefore we have identically:
d 0
f (y, Φ(y)) = 0 ,
dy t
267
A HADAMARD TYPE THEOREM FOR THE STATIC SPACE-TIME
that is, for any vector field w = w(s) having compact support and tangent to M 0
along y, for any vertical vector field τ along Φ(y), having compact support, we
have:
00
(y, Φ(y))[w, τ ] + ftt00 (y, Φ(y)) Φ0 (y)[w], τ = 0 .
ftx
(2)
£
Now recall that:
¤
f 00 (γ) [(v, τ1 ), (w, τ2 )] =
=
Z 1n
0
−
Z 1n
0
o
β
2 hDs v, Ds wiM0 − 2 hRẋ,v ẋ, wiM0 − HR
(x)[v, w] ṫ2 ds
o
2 hgrad β, viM0 ṫ τ̇2 + 2 hgrad β, wiM0 ṫ τ̇1 ds −
Z
1
0
2 β(x) τ̇1 τ̇2 ds
00
00
00
= fxx
(γ)[v, w] + fxt
(γ)[v, τ2 ] + fxt
(γ)[w, τ1 ] + ftt00 (γ)[τ1 , τ2 ]
³
´
³
´
00
00
00
= fxx
(γ)[v, w] + fxt
(γ)[w, τ1 ] + fxt
(γ)[v, τ2 ] + ftt00 (γ)[τ1 , τ2 ] ,
so that, (2) implies:
f 00 (x, Φ(x)) (v, Φ0 (x)[v]), (w, τ2 ) =
£
(3)
¤
00
00
= fxx
(x, Φ(x))[v, w] + fx,t
(x, Φ(x)) w, Φ0 (x)[v] .
£
Recall also that:
·)
··)
¤
J(x) = f (x, Φ(x)) ,
J 0 (x)[v] = fx0 (x, Φ(x))[v] + ft0 (x, Φ(x)) Φ0 (x)[v] ,
£
¤
00
00
· · ·) J 00 (x)[v, w] = fxx
(x, Φ(x)) v, Φ0 (x)[w] +
(x, Φ(x))[v, w] + fxt
£
¤
00
+ fxt
(x, Φ(x)) w, Φ0 (x)[v] + ftt00 (x, Φ(x)) Φ0 (x)[v], Φ0 (x)[w] ;
£
¤
£
¤
here, (· · ·) can be obtained easily, using a variation of x corresponding to the
directions of v and w and observing that β(x) ṫ = constant (see [7]). At this
point, the thesis springs out from (2), (3), (· · ·).
Corollary (6). The following a) and b) are equivalent.
a) J 00 (x)[v, w] = 0, for any vector field w;
b) f 00 (γ) [(v, Φ0 (x))[v], (w, τ )] = 0, for any vector field (w, τ ).
So that, the null spaces of J 00 (x) and of f 00 (γ) have the same dimension.
Proof: The equivalence a)↔b) is straightforward.
268
W. FRATTAROLO
We shall show the second statement. Let (v, τ1 ) belong to the null space of
f 00 (γ):
³
´
00
00
f 00 (γ) [(v, τ1 ), (w, τ2 )] = fxx
(γ)[v, w] + fxt
(γ)[w, τ1 ] +
³
´
00
+ fxt
(γ)[v, τ2 ] + ftt00 (γ)[τ1 , τ2 ] = 0,
So, it is identically:
(1)
(
∀ (w, τ2 ) .
00 (γ)[v, w] + f 00 (γ)[w, τ ] = 0, ∀ w,
fxx
1
xt
00 (γ)[v, τ ] + f 00 (γ)[τ , τ ] = 0,
fxt
2
1 2
tt
∀ τ2 .
Comparing the second equation of the above system (1) with the equation (2)
of the proposition (5), we obtain:
00
00
(γ)[v, τ2 ] + ftt00 (γ) Φ0 (x)[v], τ2 ,
fxt
(γ)[v, τ2 ] + ftt00 (γ)[τ1 , τ2 ] = fxt
⇒
ftt00 (γ)[τ1 , τ2 ] = ftt00 (γ) Φ0 (x)[v], τ2 ,
⇒
ftt00 (γ) τ1 − Φ0 (x)[v], τ2 = 0,
⇒
Z
1
£
£
β(x)
0
¤
¤
£
∀ τ2
¤
∀ τ2
∀ τ2
´
d³
τ1 − Φ0 (x)[v] τ̇2 ds = 0,
ds
∀ τ2 .
If (τ2 = τ1 − Φ0 (x)[v]), then:
°
Z 1°
°d ¡
¢° 2
0
°
°
° ds τ1 − Φ (x)[v] °
0
so,
ds = 0 ,
Tγ(s) M
τ1 = Φ0 (x)[v] + constant ;
but, as τ1 (0) = 0 = Φ0 (x)[v] |s=0 , then:
τ1 = Φ0 (x)[v] .
Remark (7). A vector field (v, τ ) belongs to the null space of f 00 (γ) iff it is:
(v, τ ) = v, Φ0 (x)[v]
¡
and v belongs to the null space of J 00 (x).
¢
Proof of Theorem (1): We already know, by [7], that the function expp
is onto, for any point P ∈ M . In order to prove it is one to one, it will suffice to
show that all the critical points of the functional J are minima; afterwards, the
unicity of the geodesic joining two given points on M will follow from well known
results of the critical points theory (see e.g.: [8], Theorem (6, 5, 3), page 354).
A HADAMARD TYPE THEOREM FOR THE STATIC SPACE-TIME
269
Viceversa, if expp is a diffeomorphism for any point P ∈ M , then no couple
of conjugate points with respect to the action functional exists, on any geodesic
γ contained in M ; then corollary (6) and remark (7) imply the lack of couples of
conjugate points, with respect to the functional J, on the critical curves of J.
This fact shows that expp is a diffeomorphism for any P ∈ M , iff the form
J 00 is positive definite on any critical curve of J, independently of the extreme
points of that, so Lemmas (3) and (4) complete the proof.
ACKNOWLEDGEMENTS – I would like to thank V. Benci, D. Fortunato and F. Giannoni for many enlightening discussions and the “Istituto di Matematiche Applicate” of
the faculty of engineering of the University of Pisa for its warm hospitality.
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Walter Frattarolo,
Facolta di Ingegneria, Istituto di Matematiche Applicate “U. Dini”,
Via Bonano, 25 B, I 56126 Pisa – ITALIA