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{\noindent \textsc{Proceedings of the X Fall Workshop \\
on Geometry and Physics \\ Miraflores de la Sierra, Madrid} (2001)\\
 \textsc{Publicaciones de la RSME}, Vol. 4 (2001), pp.
169-182%~\pageref{xxx}--\pageref{endxxx}.
 } \null\vskip 10mm



\pagestyle{myheadings}
\markboth
{{\small\sc Conjugate points along null geodesics}}
{{\small\sc M. Guti\'{e}rrez, F. J. Palomo and A. Romero}}



\thispagestyle{empty}

\begin{center}
{\Large\bf
Conjugate points along null geodesics on
\medskip

Lorentzian manifolds with symmetries
}
\vskip 10mm



{ \textsc{M. Guti\'{e}rrez}$^1$, \textsc{F. J. Palomo}$^1$
\textsc{and A. Romero}$^2$ } \vskip 5mm {\it $^1$Departamento de
\'{A}lgebra, Geometr\'{\i}a y Topolog\'{\i}a,\\ Universidad de
M\'{a}laga, Spain
\smallskip

$^2$Departamento de Geometr\'{\i}a y Topolog\'{\i}a,\\ Universidad
de Granada, Spain}

\smallskip

e-mails: mgl@agt.cie.uma.es, fjpalomo@eresmas.com, aromero@ugr.es

\end{center}

\bigskip


\begin{abstract}
Several integral inequalities, involving the Ricci tensor, for a
compact Lorentzian manifold which admits a timelike conformal
vector field are given. These inequalities relate conjugate points
along null geodesics to global geometric properties. As a
consequence, some classification results on certain compact
Lorentzian manifolds without conjugate points along its null
geodesics are shown.

\bigskip
\it \noindent {\rm Key words:} Lorentzian manifolds, timelike conformal vector
fields, null geodesics, conjugate points.

\noindent {\rm MSC 2000:} 53C50, 53C22.
\end{abstract}

\hyphenation{res-pec-ti-ve-ly}
\hyphenation{Re-la-ti-vi-ty}
\hyphenation{ma-ni-fold}
\hyphenation{cons-tant}
\hyphenation{glo-ba-lly}
\hyphenation{res-trict}
\hyphenation{e-ve-ry}
\hyphenation{nor-ma-li-zed}

\section{Introduction}

On a Lorentzian manifold $(M,g)$, tangent vectors are classified into {\it timelike,
null,} or {\it spacelike}, and so a (smooth) curve on $M$ is said to be timelike,
null, or spacelike if its tangent vectors are always timelike, null, or spacelike,
respectively. Lorentzian Geometry is the mathematical theory used in General
Relativity. In fact, a timelike curve corresponds to the path of an observer moving
at less than the speed of light. Null curves correspond to moving at the speed of
light, and spacelike curves to moving faster than light. Although Relativity
predicts that physical particles cannot move faster than light, spacelike curves are
of undeniable geometric interest. If a timelike or null curve is supposed to be a
geodesic, then it represents the path of a ``free falling'' particle or the path of
a lightlike particle, respectively.

It is commonly argued in General Relativity that the curvature tensor codifies the
gravity, and the relative position of neighbours of a free falling particle $\gamma
$, is given by the \textit{Jacobi fields} on $\gamma $. In this way, if the gravity
attracts then it will cause conjugate points while if gravity does not attract then
it will prevent them. From a geometric point of view, a conjugate point $\gamma (a)$
of $ p=\gamma (0)$ along a geodesic $\gamma $ can be interpreted as an
``almost-meeting point'' of geodesics starting from $p$ with initial velocities near
$\gamma ^{\prime }(0)$. These neighbouring geodesics may, but need not, actually
pass through the point $\gamma (a)$. The existence of conjugate points on timelike
geodesics gives an effect rather like the ``twin paradox'' and for null geodesics,
may be related to the phenomenon of gravitational lensing \cite{Hel94}. Moreover,
the study of conjugate points along causal geodesics plays an important role to
state Singularity theorems and to study Causality Theory, see \cite{One83} and
\cite{HawEll73}, for instance.

We will be interested here in compact Lorentzian manifolds. All these manifolds are
acausal, i.e. they possess closed timelike curves \cite[Lemma 14.10] {One83}, while
a physically admissible spacetime is generally assumed to be free of closed causal
curves. Yurtsever introduced \cite{Yurt90} a new notion of causality for general
acausal Lorentzian manifolds. Recall that remarkable examples of spacetimes $M$ are
often merely a connected open set in a larger spacetime $\widetilde{M}$, and $M$ can
be causal even though $\widetilde{M}$ is not \cite{One95}. Moreover, it has been
pointed out \cite{New86} that general properties of compact spacetimes should be
identified in order to show they are incompatible with astronomical observations,
since the objections to chronology violations are based more on philosophy than
physics. On the other hand, following \cite{Yurt90}, it is reasonable to expect that
Lorentzian theory on a compact manifold will provide valuable information on the
topology of the underlying manifold, complementary to the one obtained through the
study of Riemannian theory.

It is well-known that not every compact manifold, $M$, can be endowed with a
Lorentzian metric. This holds if and only if the Euler number of $M$ vanishes
\cite[Prop. 5.57]{One83}. Moreover, for Lorentzian metrics, compactness does not
imply geodesic completeness, in fact the \textit{Clifton-Pohl torus} is an example
of this situation \cite[Ex. 7.16]{One83}, \cite{RomSan94b}. Of course, this is not
an isolated example. A general procedure to construct many incomplete Lorentzian
metrics on the torus was explained in \cite{RomSan94}. On the other hand, geodesic
completeness in the Lorentzian setting can be separated into spacelike, null and
timelike completeness. These conditions are shown to be independent  \cite[Ex.
5.43]{One83}, \cite{RomSan94b} and references therein. See also the comment at the
end of this paper.

A way to get completeness in the compact Lorentzian case is to impose some geometric
extra condition on the manifold \cite{RomSan94b}. In fact, it is relevant for this
work that the existence of a timelike conformal vector field on a compact Lorentzian
manifold yields to its geodesic completeness  \cite{RomSan95}.

A compact Riemannian manifold must be complete as an easy consequence of the
classical Hopf-Rinow theorem. Another argument to obtain the same conclusion is
\cite[p. 32]{Bes78}: take the unit tangent bundle $UB$ of an $n-$dimensional
Riemannian manifold $(B,g_B)$. It is a fiber bundle over $B$ with fiber type the
$(n-1)-$dimensional unit sphere $\S ^{n-1}$. If $B$ is assumed to be compact, then
$UB$ is also compact and therefore the geodesic flow is automatically complete as it
is a flow of a vector field on a compact manifold. This argument fails in the
Lorentzian setting because there is no reasonable fiber bundle over a Lorentzian
manifold $(M,g)$ with compact fiber and having a tangent vector for every direction
in $T_{p}M$ for all $p\in M$. In fact, if we put $\left. U^{+}M=\left\{ v\in
TM:g(v,v)=+1\right\}\right. $ and $U^{-}M=\left\{ v\in TM:g(v,v)=-1\right\}$, then
both fiber bundles have not compact fibers and neither of them has a vector for
every direction of $TM$. In order to avoid this difficulty, we will restrict our
attention to null tangent directions on a Lorentzian manifold which admits a
timelike conformal vector field. Thus the key tool in our work is the \textit{null
congruence associated with a timelike conformal vector field }$K$ which can be seen
as the manifold of all null tangent directions of $(M,g)$, and it is denoted by
$C_{K}M$. This null congruence is a fiber bundle over $M$ with compact fiber type
$\S ^{n-2}$. As we will point out (see section 4) the geodesic flow preserves
$C_{K}M$. So, a reasoning as in the previous Riemannian case gives the null
completeness of $M$, already known from \cite{RomSan95}.

The main aim of this note is to introduce, following \cite{GutPal Rom01} and
\cite{GutPalRom02}, two \textit{new integral inequalities} on compact Lorentzian
manifolds which admit a timelike conformal vector field. These integral inequalities
relate conjugate points along null geodesics with global properties of the
Lorentzian manifold. In particular, we have found an ``splitting'' result (Corollary
\ref{Corollary2}) and several classification results on certain compact Lorentzian
manifolds without conjugate points on their null geodesics.

The study of conformal vector fields on pseudo-Riemannian geometry is a topic of
special importance, see for instance \cite{KühRad97}. In particular, the study of
timelike conformal vector fields on Lorentzian Geometry has been developed mainly
under assumptions of interest in physics and, from a mathematical viewpoint, the
case of timelike Killing vector fields has appeared to be a useful tool to get
classification theorems in some area of Lorentzian Geometry \cite{San97}. Moreover,
standard Lorentzian space forms (those with a time-orientable double Lorentzian
covering admitting a timelike Killing vector field) have been studied in \cite
{KulRay85}. Recall furthermore, that the existence of a timelike Killing vector
field has been also used to classify compact Lorentzian space forms \cite {Kam93}
and more generally to study and classify compact Einstein Lorentzian manifolds
\cite{RomSan96}, \cite{RomSan98}. Finally, note that the existence of a timelike
conformal vector field has been also used to study compact spacelike hypersurfaces
of constant mean curvature (see \cite{AliRomSan97}, \cite{AliRomSan00} and
references therein).




\section{Motivations from the Riemannian case}

In 1948, E. Hopf published a remarkable theorem where he showed that {\it the total
scalar curvature of a closed surface without conjugate points is nonpositive and
vanishes only if the surface is flat}. Therefore, thanks to the Gauss-Bonnet
formula, a Riemannian torus without conjugate points is flat \cite{Hopf48}. Hopf's
result was, ten years after, extended by L.W. Green \cite{Gre58} to compact
Riemannian manifolds as follows: \textit{if }$(B,g_{B})$ \textit{has no conjugate
points then }
\begin{equation}
\int_{B}Sd\mu _{g_{B}}\leq 0,  \label{hopf-green}
\end{equation}
\textit{and it vanishes only if} \textit{the metric is flat,} (here $S$ and $d\mu
_{g_{B}}$ denote, respectively, the scalar curvature of $(B,g_{B})$ and the
canonical measure associated to $g_{B}$). More recently, F. Guimaraes, generalized
in \cite{Gui92} the Green theorem to complete Riemannian manifolds under the
assumption that the Ricci curvature on the unit tangent bundle has an integrable
positive or negative part.

On the other hand, in  \cite{Gre63} L. W. Green solved the famous Blaschke
conjecture in dimension two. A useful tool to prove it was the so-called Berger
inequality which asserts that $\mathrm{area}(B,g_{B})\geq \frac{2a^{2}}{\pi }\chi
(B)$, being $\chi (B)$ the Euler number of $B$, for a $2-$dimensional compact
Riemannian manifold without conjugate points before a fixed distance $a$ in the
parameter of any (unit) geodesic. Moreover, the equality holds only if $(B,g_{B})$
has constant sectional curvature $\frac{\pi ^{2}}{a^{2}}$. This inequality was later
generalized to higher dimensions by Berger and independently by Green \cite[Prop.
5.64]{Bes78} as follows:
\begin{equation}
\mathrm{Vol}(B,g_{B})\geq \frac{a^{2}}{\pi ^{2}n(n-1)}\int_{B}Sd\mu _{g_{B}},
\label{berger-green}
\end{equation}
where $(B,g_{B})$ is a compact Riemannian manifold of dimension $n$ without
conjugate points before a fixed distance $a$ in the parameter of any geodesic.
Moreover, the equality holds if and only if $(B,g_{B})$ has constant sectional
curvature $\frac{\pi ^{2}}{a^{2}}$. Note that Berger's inequality is a direct
consequence of (\ref{berger-green}), via the Gauss-Bonnet formula.

Inspired from these results, we want to study Lorentzian manifolds in which no null
geodesic contains a pair of mutually conjugate points before a fixed value of a
special affine parameter. In fact, assuming the existence of a timelike conformal
vector field, we will distinguish an affine parameter for each null geodesic, and we
deduce an integral inequality in the same philosophy to (\ref{berger-green}). We
will explote several geometric consequences and show that the Hopf-Green and the
Berger-Green inequalities above can be deduced as a particular case of our approach.

\section{Setup}

Let $(M,g)$ be an $n(\geq 2)-$dimensional Lorentzian manifold; that is a (connected)
smooth manifold $M$ endowed with a non-degenerate metric $g$ with signature
($-,+,.....,+).$ We denote $TM$ for the tangent fibre bundle of $M$, $\pi
:TM\longrightarrow M$ for the natural projection and for every $ v\in TM$, we write
$(\ )_{v}:T_{\pi (v)}M\longrightarrow T_{v}T_{\pi (v)}M$ for the natural
identification. We shall write $\nabla $ for its Levi-Civita connection, $R$ for its
Riemannian curvature tensor (our convention on the curvature tensor is
$R(X,Y)Z=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z$),
$\mathrm{Ric}$ for its Ricci tensor, $S$ for its scalar curvature and $d\mu _{g}$
for the canonical measure associated with $ g $.

As usual, the causal character of a tangent vector $v\in T_{p}M$ is timelike (resp.
null, spacelike) if $g(v,v)<0$ (resp. $g(v,v)=0$ and $v\neq 0,$ $ g(v,v)>0$ or
$v=0)$. If $v\in T_{p}M$ then, $\gamma _{v}$ will denote the unique geodesic such
that $\gamma _{v}(0)=p$ and $\gamma _{v}^{\prime }(0)=v$ . It is well-known that the
causal character of $\gamma ^{\prime }(t)$, for any geodesic $\gamma $ of $(M,g)$,
does not depend on the parameter $t$. In particular, a null geodesic $\gamma $ of
$(M,g)$ is a geodesic such that $ \gamma ^{\prime }(t)$ is a null vector. A vector
field $K\in \mathfrak{X}(M)$ is said to be timelike if $K_{x}$ is timelike for all
$x\in M$.

Recall that the cotangent bundle $T^{*}M$, of an arbitrary manifold $M$, carries a
natural symplectic structure given by $d\alpha $ where $\alpha $ is the 1-form
defined by $\alpha (\xi )=-q(\xi )\left[ p_{*}(\xi )\right] $
for all $\xi \in TT^{*}M$, being $q:TT^{*}M\longrightarrow T^{*}M$, and $%
p:T^{*}M\longrightarrow M$ the natural projections. If $(M,g)$ is a Lorentzian
manifold, then $g$ determines a vector bundle isomorphism from $TM$ to $T^{*}M$,
$\left. \flat :TM\longrightarrow T^{*}M \right. $, by putting $v\mapsto g(v,\ )$. If
we call $\alpha _{g}$ the pull-back of $\alpha $ by $\flat $, then $\left. \alpha
_{g}(X)=-g(v,\pi _{*}(X)) \right. $, where $X\in T_{v}(TM)$. The geodesic vector
field, $Z_{g}$, is the vector field on $TM$ defined by $ i_{Z_{g}}d\alpha _{g}=dE$,
where $E$ is given by $E(v)= \frac{1}{2}g(v,v)$. The flow $\left\{ \Phi _{t}\right\}
$ of $Z_{g}$, $\Phi _{t}(v)=\gamma _{v}^{\prime }(t)$, is called the
\textit{geodesic flow} of $ (M,g)$.

We point out that the {\it Sasaki metric} \, $\widehat{g}$ \, induced from the
Lorentzian metric $g$ may be introduced on $TM$ in a similar way to the Riemannian
case. Note that now $(TM,\widehat{g})$ is a semi-Riemannian manifold with index two,
and $\pi :(TM,\widehat{g} )\longrightarrow (M,g)$ is a semi-Riemannian submersion.

As it has been said above, we are here interested in the study of conjugate points
along null geodesics (or null conjugate points). Many authors have thought about
this topic, we will quote here several significant results:

\begin{enumerate}
\item  A conjugate point along a null geodesic $\gamma $ is an almost-meeting point
of nearby null geodesics, i.e. the geodesic variation of $\gamma $ could be chosen
such that all its longitudinal geodesics are null geodesics \cite[p. 291]{One83}.
This fact is not obvious from continuity arguments.

\item  An essential problem in Lorentzian Geometry is to determine if a given pair
of points can be joined by a timelike curve (Causality Theory). This is related to
conjugate points along null geodesics as follows: if $ \gamma :[0,a]\longrightarrow
M$ is a null geodesic with $\gamma (0)$ and $ \gamma (a)$ conjugate points along
$\gamma $, then for every $\varepsilon >0$ there is a timelike curve from $\gamma
(0)$ to $\gamma (a+\varepsilon )$ ``arbitrarily near'' to $\gamma $ \cite[Prop.
10.48]{One83}.

\item  Another important question is to find incomplete geodesics (Singularity
Theory). The \textit{Hawking-Penrose conjugancy} theorem for null geodesics connects
Singularity Theory to null conjugate points, and it can be stated as follows: if
$\gamma _{v}:I\longrightarrow M$ is a null geodesic, $\mathrm{Ric}(\gamma
_{v}^{\prime },\gamma _{v}^{\prime })\geq 0$, and $R(\ ,\gamma _{v}^{\prime
}(t))\gamma _{v}^{\prime }(t)$, contemplated as a linear operator of the quotient
space $\mathrm{Span}\{ \gamma _{v}^{\prime }(t)\} ^{\perp }\diagup \mathrm{Span}\{
\gamma _{v}^{\prime }(t)\}$, is not the zero operator for some $t\in I$, then $
\gamma _{v}$ is incomplete or has a pair of conjugate points (see for instance
\cite[Prop. 12.17]{BeeEhrEas96}).

\item  No null geodesic $\gamma $ in any two-dimensional Lorentzian manifold has
conjugate points \cite[Lemma 10.45]{BeeEhrEas96}.

\item  There is no conjugate point along null geodesics in any Lorentzian manifold
of constant sectional curvature \cite[Ex. 10-11]{One83}. Nevertheless, it is easy to
construct examples of Lorentzian manifolds without conjugate points along their null
geodesics and which do not have constant sectional curvature.
\end{enumerate}

Of course, sectional curvature can be defined on Lorentzian manifolds, but it plays
a different role than in the Riemannian case. In fact, it cannot be stated for null
planes (degenerate planes in the induced metric). Now, we close this section by
pointing out two of these differences.

\begin{enumerate}
\item  There is no compact Lorentzian manifold of constant sectional curvature
$c>0$. In fact, for $n=2$, it can be directly deduced from the Gauss-Bonnet formula
for Lorentzian metrics. For $n\geq 3$, this is a consequence of a classical result
by Calabi and Markus \cite{CalMar62} which implies the inexistence of compact
quotients of \textit{de Sitter space,} and a more recent result of completeness by
Klinger \cite{Kli96}.

\item  In 1979, Kulkarni showed that if a (connected) $n(\geq 3)-$dimensional
Lorentzian manifold $(M,g)$ has sectional curvature bounded from above or bounded
from below, then $(M,g)$ has constant sectional curvature \cite{Kul79} (see also
\cite[Prop. 8.28]{One83}).
\end{enumerate}

We end this section by recalling the well-known notion of {\it null sectional
curvature} \cite{Har82}, \cite{Har85}, \cite[Definition A.6]{BeeEhrEas96} to be used
later. If $v$ is a null vector and  $\sigma $ a null plane containing it, the null
sectional curvature with respect to $v$ of the plane $\sigma $ is defined to be
$\mathcal{K} _{v}(\sigma )=g(R(u,v)v,u)/g(u,u)$, where $u(\neq 0)$ is any vector in
$\sigma $ independent with $v$ (and therefore spacelike). This curvature does not
depend on the choice of the non-zero spacelike vector $u,$ but it does quadratically
on $v$. An $n(\geq 3)-$dimensional Lorentzian manifold has constant sectional
curvature if and only if has null sectional curvature everywhere zero \cite[Prop.
8.28]{One83}.



\section{Null Congruence}


 From now on $(M,g)$ will be a (time orientable) Lorentzian manifold with dimension
$n\geq 3$, and $K$ a timelike vector field on $M$.  Recall \cite{Har85},
\cite{Koc85} that the \textit{null congruence associated to }$K$ is defined by

\begin{equation}
C_{K}M=\left\{ v\in TM:g(v,v)=0, \ g(v,K_{\pi (v)})=1\right\}.
\end{equation}

The null congruence has the following nice properties: for each null tangent vector
$v,$ there exists a unique $t\in \R$ such that $tv\in C_{K}M$, and the map
$v\longmapsto \left[ v\right] $ is a diffeomorphism from $C_{K}M$ to the manifold of
the null directions of $M$, $\mathcal{N}=\left\{ \left[ v\right] \in
PM:g(v,v)=0\right\}$ (here we denote by $PM$ the projective fiber bundle associated
to $TM$).

The null congruence $C_{K}M$ is an orientable embedded submanifold of $TM$ with
dimension $2(n-1)$. Moreover $(C_{K}M,\pi ,M)$ is a fiber bundle with fibre type $\S
^{n-2}$, and so $C_{K}M$ will be compact if $M$ is compact. If $\widehat{g}$ also
represents the induced metric on $C_{K}M$ from the Sasaki one of $TM$, then
$(C_{K}M,\widehat{g})$ is a Lorentzian manifold and the restriction of $\pi $ to
$C_{K}M$ is a semi-Riemannian submersion with spacelike fibers (see \cite{GutPal
Rom01} for details). We shall write $d\mu _{\widehat{g}}$ for the canonical measure
on $C_{K}M$ induced from $\widehat{g}$.

This manifold is our main tool in the study of null conjugate points. \relax From a
geometric point of view, the null congruence has been systematically studied in
\cite{GutPal Rom01}, where the reader can find the following results:

\begin{enumerate}

\item  The volume element of $(C_{K}M,\widehat{g})$ is given, up to a constant
multiple, by the $2(n-1)-$form $\beta _{g}\wedge \alpha _{g}\wedge (d\alpha
_{g})^{n-2},$ where $\beta _{g}(X)=-g(K_{\pi v},\pi _{*}(X))$ for every $X\in
T_{v}C_{K}M$.

\item  The fiber $\pi ^{-1}(x)=\left( C_{K}M\right) _{x}$ is isometric to $ \left(
\S ^{n-2},-g(K_{x},K_{x})^{-1}g_{can}\right) $, where $\left( \S
^{n-2},g_{can}\right) $ is the canonical unit Riemannian sphere. Therefore $\left(
C_{K}M\right) _{x}$ is isometric to a Riemannian sphere of radius $\left[
-g(K_{x},K_{x})\right] ^{-1/2}$. Note that there is a remarkable difference with the
unit tangent bundle of a Riemannian manifold, where all the fibers are isometric. We
will write $h=\left[ -g(K,K)\right] ^{-1/2}$ and so $g(U,U)=-1$ with $U=hK$.

\item
$C_{K}M$ is invariant by the geodesic flow if and only if
the vector field $K$ is assumed to be conformal, i.e. $\mathfrak{L}_{K}g=\rho g$ for
some $\rho \in C^{\infty }(M)$. Moreover, in such a case $\mathrm{div}\left(
Z_{g}\mid _{C_{K}M}\right) =0,$ where $ \mathrm{div}$\textrm{\ }denotes the
divergence operator of $(C_{K}M,\widehat{ g})$. So, if $(M,g)$ is compact and $K$ is
conformal (therefore $(M,g)$ is geodesically complete \cite{RomSan95}) then, we
have:
\begin{equation}
\int_{C_{K}M}(f\circ \Phi _{t})d\mu _{\widehat{g}}=\int_{C_{K}M}fd\mu _{
\widehat{g}}  \label{Liouville}
\end{equation}
for every $f\in C^{0}(C_{K}M)$ and $t\in \R$.

\item  If $K$ is conformal, then every null geodesic $\gamma _{v}$ of $(M,g)$ with
$v\in C_{K}M$, gives rise to the null geodesic $\gamma _{v}^{\prime }$ of
$(C_{K}M,\widehat{g})$. Furthermore, each null geodesic $\beta $ of $ (M,g) $ may be
reparametrized to obtain a null geodesic $\alpha $ which satisfies $\alpha ^{\prime
}(t)\in C_{K}M.$ In fact, $g(\beta ^{\prime },K_{\beta })=a\in \R,$ $a\neq 0.$ Thus,
if we put $\alpha (t)=\beta ( \frac{t}{a})$ we achieve $g(\alpha ^{\prime
},K_{\alpha })=1$.
\end{enumerate}

If a null congruence associated with a timelike vector field $K$ (conformal or not)
has been fixed, then we may choose, for every null plane $\sigma $, the unique null
vector $v\in C_{K}M\cap \sigma $. Therefore, the null sectional curvature can be
thought as a function on null planes. Along this note we always use such convention,
and we will call it the $K-$\textit{normalized null sectional curvature}.

\section{The integral inequalities}

We recall that if $\gamma _{v}:[0,a]\longrightarrow M$ is a null geodesic such that
there are no conjugate points of $\gamma _{v}(0)$ in $[0,a)$, then the
\textit{Hessian form }$H_{\gamma _{v}}^{\perp }$ is positive semidefinite, i.e.
\begin{equation}
H_{\gamma _{v}}^{\perp }(V,V):=\int_{0}^{a}\left[ g\Big(\frac{\nabla V}{dt},\frac{
\nabla V}{dt}\Big)-g(R(V,\gamma _{v}^{\prime })\gamma _{v}^{\prime },V)\right]
dt\geq 0,  \label{Hessiano}
\end{equation}
for every vector field $V$ along $\gamma _{v}$ such that $ V(0)=0$,
$\left. V(a)=0 \right. $ and $g(\gamma _{v}^{\prime },V)=0,$ \cite[pp.
290-1]{One83}. Moreover, a standard argument permits us to show, that if $H_{\gamma
_{v}}^{\perp }(V,V)=0$ then $\frac{\nabla ^{2}V}{
dt^{2}}+R(V,\gamma _{v}^{\prime })\gamma _{v}^{\prime }=f \gamma _{v}^{\prime }$
where $f$ is a smooth function. Observe that, in contrast to the well-known
Riemannian case, here the condition $H_{\gamma _{v}}^{\perp }(V,V)=0$ does not
imply, in general, that $V$ is a Jacobi vector field.

    From now on, we will assume that $M$ is (($n \geq 3$)-dimensional) compact and
$(M,g)$ admits a timelike conformal vector field $K$.

We are ready to introduce the integral inequalities that involve null conjugate
points. The proof of the following result makes use of (\ref {Liouville}) and
(\ref{Hessiano}). It can be found in \cite[Th. 3.2] {GutPal Rom01}. We put
$\widehat{\rm{Ric}}(v)={\rm Ric}(v,v)$ for all $v \in C_{K}(M)$.

\begin{theorem}
\label{Theorem2}If there exists $a\in (0,+\infty )$ such that every null geodesic
$\gamma _{v}:\left[ 0,a\right] \longrightarrow M$, with $ v\in C_{K}M$, has no
conjugate point of $\gamma _{v}(0)$ in $[0,a)$, then
\begin{equation}
\mathrm{Vol}(C_{K}M,\widehat{g})\geq \frac{a^{2}}{\pi ^{2}(n-2)}
\int_{C_{K}M}\widehat{\rm{Ric}} \ d\mu _{\widehat{g}}.  \label{13}
\end{equation}
Moreover, equality holds if and only if $(M,g)$ has constant $K-$normalized null
sectional curvature $\frac{\pi ^{2}}{a^{2}}$.
\end{theorem}

Notice that the equality holds in (\ref{13}) if and only if $(M,g)$ has
$U-$normalized null sectional curvature $-\frac{\pi ^{2}}{a^{2}}g(K,K)$. That is,
the $U-$normalized null sectional curvature of $(M,g)$ is an everywhere non-zero
point function. An equivalent condition to this \cite[Remark 3.4]{GutPal Rom01} was
studied by H. Karcher, \cite{Kar82}. He showed, when $n\geq 4$, the following
theorem:

\begin{quotation}
\textit{Let} $U$ \textit{be a unit timelike vector field on an }$n(\geq 4)-$
\textit{dimensional Lorentzian manifold $(M,g)$. The $U-$normalized null sectional
curvature is a everywhere non-zero point function
if and only if:} \\
\textit{1. The distribution }$U^{\perp }$\textit{\ is integrable.} \\
\textit{2. The integral manifolds of }$U^{\perp }$\textit{\ are totally
umbilic and have constant sectional curvature.} \\
\textit{3. }$(M,g)$\textit{\ is locally conformal to a flat Lorentzian space.
}
\end{quotation}

We would like to point out that conclusion 1 above does not remain true if it
is assumed $\dim M=3$ \cite[Remark 4.4]{GutPal Rom01}.

On the other hand, as it has been proved in \cite{GutPal Rom01}:
\begin{equation}
\mathrm{Vol}(C_{K}M,\widehat{g})=\mathrm{Vol}(\S ^{n-2},g_{can})
\int_{M}h^{n-2}d\mu _{g}
\end{equation}
and
\begin{equation}
\int_{C_{K}M}\widehat{\rm {Ric}} \ d\mu _{\widehat{g}}=\frac{\textrm{Vol}(
\S ^{n-2},g_{can})}{n-1}\int_{M}\left[ n\mathrm{Ric}(U,U)+S\right]
h^{n}d\mu _{g}.
\end{equation}

Therefore, we can rewrite Theorem \ref{Theorem2} as follows:

\begin{theorem}
\label{Theorem3}If there exists $a\in (0,+\infty )$ such that every null geodesic
$\gamma _{v}:\left[ 0,a\right] \longrightarrow M$, with $ v\in C_{K}M$, has no
conjugate point of $\gamma _{v}(0)$ in $[0,a)$, then
\begin{equation}
\int_{M}h^{n-2}d\mu _{g}\geq \frac{a^{2}}{\pi ^{2}(n-1)(n-2)}\int_{M}\left[ n
\mathrm{Ric}(U,U)+S\right] h^{n}d\mu _{g}.  \label{30}
\end{equation}
Moreover, equality holds if and only if $(M,g)$ has $U-$normalized null sectional
curvature $-\frac{\pi ^{2}}{a^{2}}g(K,K)$.
\end{theorem}

If any null geodesic does not contain a pair of mutually conjugate points, then
(\ref{30}) is valid for all positive $a$. Since $\int_{M}h^{n-2}d\mu _{g}<+\infty $,
we achieve (\ref{Hopf2}). The equality case in (\ref{Hopf2}) requires a more
intricate argument \cite{GutPalRom02}.

\begin{theorem}
\label{Teorema1}If there are no conjugate points along the null geodesics, then
\begin{equation}
\int_{M}\left[ n\mathrm{Ric}(U,U)+S\right] h^{n}d\mu _{g}\leq 0.
\label{Hopf2}
\end{equation}
Moreover, equality holds if and only if $(M,g)$ has constant sectional curvature
$c\leq 0$.
\end{theorem}

This result allows us to show the existence of null conjugate points without using
the Jacobi equation \cite[Cor. 4.3]{GutPal Rom01}. Moreover, it can be thought as an
integral condition to obtain constant sectional curvature from the absence of null
conjugate points (compare with \cite {EhrKim94}).

If $K$ is assumed, more restrictibly, to be \textit{Killing}, then we have proved in
\cite[Lemma 3.12]{GutPal Rom01} that $\int_{M}\mathrm{Ric}(U,U)h^{n}d\mu _{g}\geq
0$, and equality holds if and only if $U$ is parallel (and hence, $h$ is a constant
function). This fact permits us to improve, in that case, Theorems \ref{Theorem3}
and \ref{Teorema1} to get:

\begin{corollary}
\label{Corollary2}If $K$ is Killing and there is $a\in (0,+\infty )$ such that every
null geodesic $\gamma _{v}:\left[ 0,a\right] \longrightarrow M$, $v\in C_{K}M$, has
no conjugate point of $\gamma _{v}(0)$ in $[0,a)$, then
\begin{equation}
\int_{M}h^{n-2}d\mu _{g}\geq \frac{a^{2}}{\pi ^{2}(n-1)(n-2)}
\int_{M}Sh^{n}d\mu _{g}.  \label{36}
\end{equation}
Moreover equality holds if and only if $g(K,K)$ is constant and the universal covering of $(M,g)$ is globally
isometric to the semi-Riemannian product \\ $\Big( \R \times \S ^{n-1}$($\frac{ah}{\pi
}),-dt^{2}+g_{can} \Big)$.
\end{corollary}

\begin{corollary}
\label{corollary1}If $K$ is Killing and there are no conjugate points along the null
geodesics, then
\begin{equation}
\int_{M}Sh^{n}d\mu _{g}\leq 0.  \label{desigualdad3}
\end{equation}
The equality holds if and only if $(M,g)$ is flat. Moreover, in this case, $ U $ is
parallel, the first Betti number of $M$ is not zero and the Levi-Civita connection
of $g$ is Riemannian.
\end{corollary}

\begin{remark}
\label{re1}{\em Kamishima proved that if a compact Lorentzian manifold admits a
timelike Killing vector field and has constant sectional curvature $c$, then it is
complete and $c\leq 0$. Moreover, it is affinely diffeomorphic to a Riemannian
manifold with non-zero first Betti number if $c=0$ \cite[Th. A (1)]{Kam93}. Last
result was extended in \cite[Th. 3.2]{RomSan96} to a compact Ricci-flat Lorentzian
manifold which admits a timelike Killing vector field, and later, in \cite[Cor.
3.9]{RomSan98}, to the conformal case. Since a Lorentzian manifold of constant
sectional curvature has no conjugate point along its null geodesics, Corollary
\ref{corollary1} is a proper extension (in other direction to \cite[Cor.
3.9]{RomSan98}) of Kamishima's theorem.}
\end{remark}

Note that in case that the equality holds in Corollary \ref{corollary1}, from
\cite[Cor. 3.4.6]{Wol79}, there is a (finite) Lorentzian covering of $(M,g)$ by a
flat Lorentzian torus \cite{GutPalRom02}.

Corollaries \ref{Corollary2} and \ref{corollary1} contain, as a very particular
case, the classical Berger-Green \cite{Gre63} and Hopf-Green \cite {Gre58}
inequalities in Riemannian geometry, respectively.

\begin{corollary}
Let $(B,g_{B})$ be a compact Riemannian manifold of dimension \linebreak $ n\geq 2$
and scalar curvature $S$. Suppose that no unit geodesic $\gamma
:[0,a]\longrightarrow B$ has a conjugate point of $\gamma (0)$ in $[0,a)$, then

\begin{equation}
{\rm Vol}(B,g_{B})\geq \frac{a^{2}}{\pi ^{2}n(n-1)}\int_{B}Sd\mu
_{g_{B}}.  \label{36.1}
\end{equation}
Moreover, equality holds if and only if $(B,g_{B})$ has constant sectional curvature
$\frac{\pi ^{2}}{a^{2}}$.
\end{corollary}

\noindent {\it Proof.} If we take $(M,g)=(\S ^{1}\times B,-g_{can}+g_{B})$ and the
unit timelike Killing vector field $K$ as the lift to $\S ^{1}\times B$ of the
vector field $z\longmapsto iz$ on $\S ^{1} \subset \C$, then it is easy to see that
every null geodesic $\gamma _{v}$ of $(M,g)$, with $v\in C_{K}M$, has no conjugate
point in $[0,a)$. Therefore Corollary \ref{Corollary2} gives
\begin{equation}
\mathrm{Vol}(M,g)\geq \frac{a^{2}}{\pi ^{2}n(n-1)}\int_{M}S_{g}d\mu _{g},
\label{1}
\end{equation}
where $S_{g}$ denotes the scalar curvature of $(M,g)$, and equality holds if and
only if $(M,g)$ has $K-$normalized null sectional curvature $\pi ^{2}\diagup a^{2}$.
Taking into account that \textrm{Vol}$ (M,g)=2\pi $\textrm{Vol}$(B,g_{B})$ and
$S_{g}(z,x)=S(x)$ for $ \left. (z,x)\in M \right. $, we obtain (\ref{36.1}) from
(\ref{1}). Finally note that $(M,g)$ has $K-$normalized null sectional curvature
$\pi ^{2}\diagup a^{2}$ if and only if $(B,g_{B})$ has constant sectional curvature
$\pi ^{2}\diagup a^{2}$. $\blacksquare$

\begin{corollary}
Let $(B,g_{B})$ be a compact Riemannian manifold of dimension $n\geq 2$ and scalar
curvature $S$, if $(B,g_{B})$ has no conjugate point then
\begin{equation}
\int_{B}Sd\mu _{g_{B}}\leq 0.  \label{3}
\end{equation}
Moreover, equality holds if and only if $(B,g_{B})$ is flat.
\end{corollary}

\noindent {\it Proof.} It is easy to see that no null geodesic $\gamma _{v}$ of
$(M,g)$ has conjugate points. Therefore Corollary \ref{corollary1} gives
\begin{equation}
\int_{M}S_{g}d\mu _{g}\leq 0,  \label{2}
\end{equation}
and equality holds if and only if $(M,g)$ is flat. Then we get (\ref{3}) from
(\ref{2}). Finally $(M,g)$ is flat if and only if $(B,g_{B})$ is flat.
$\blacksquare$

Note that, in the Lorentzian case, only null geodesics are involved in the
assumptions of the integral inequalities, whereas in the Riemannian setting, all
geodesics are implicated. Nevertheless, Theorem \ref{Theorem3} and Corollary
\ref{Corollary2} give us, in particular, information about the \textit{volume} of
the Lorentzian manifold $(M,g)$ only from the study of its null conjugate points. In
fact, compactness of $M$ implies that there exists an upper bound $b>0$ for the
function $h$, and so we have $ \int_{M}h^{n-2}d\mu _{g}\leq
b^{n-2}\mathrm{Vol}(M,g)$.

Another relationships of null geodesics to global properties of Lorentzian manifolds
have been pointed out in previous works. For instance, in \cite {RomSan93} it was
conjectured that a null complete compact Lorentzian manifold must be complete.
Although this conjecture has deserved the attention of some geometers
\cite{CarRoz94}, it remains now as an open problem.

\vspace{-2mm}

\section*{Acknowledgments}

The first author has been partially supported by MCYT-FEDER Grant BFM2001-1825 and
CECJA FQM-803, the second author has been partially supported by CECJA FQM-803, and
the third author has been partially supported by MCYT-FEDER Grant
BFM2001-2871-C04-01.


\hyphenation{Re-la-ti-vi-ty}
\hyphenation{ma-ni-folds}
\hyphenation{Geo-me-try}

\vspace{-3mm}


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