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\begin{document}
\title{A Berger-Green type inequality for compact Lorentzian manifolds}
\author{Manuel Guti\'{e}rrez, Francisco J. Palomo}
\author{Alfonso Romero}
\subjclass{53C50, 53C22.}
\thanks{The first author has been partially supported by DGICYT Grant BFM2001-1825, and
the third has been partially supported by DGICYT Grant BFM2001-2871-C04-01.}
\keywords{Lorentzian manifolds, null geodesics, conjugate points, Lorentzian odd
dimensional spheres.}
\address{Manuel Guti\'{e}rrez, Francisco J. Palomo: Departamento de \'{A}lgebra,
Geometr\'{i}a y Topolog\'{i}a. Facultad de Ciencias, Universidad de
M\'{a}laga, Campus Teatinos, 29071- M\'{a}laga, Spain.}
\email{M.G: {\it mgl@agt.cie.uma.es}\\
F.J.P: {\it fpalo1@clientes.unicaja.es}}
\address{Alfonso Romero: Departamento de Geometr\'{i}a y Topolog\'{i}a, Universidad
de Granada, 18071-Granada, Spain.}
\email{{\it aromero@goliat.ugr.es}}

\begin{abstract}
We give a Lorentzian metric on the null congruence associated with a
timelike conformal vector field. A Liouville type theorem is proved and a
boundeness for the volume of a compact Lorentzian manifold which admits a
timelike conformal vector field, analogous to a well known Berger-Green
theorem in the Riemannian case, will be derived by studying conjugate points
along the null geodesics. As a consequence, several classification results on
certain compact Lorentzian manifolds without conjugate points on its null
geodesics are obtained. Finally, some properties of null geodesics of some
natural Lorentzian metrics on odd dimensional spheres have been found.
\end{abstract}

\maketitle


\section{INTRODUCTION}

In the paper ''Auf wiedersehensfl\"{a}chen'' \cite{Gre63} L. W. Green solved
the famous Blaschke conjecture in dimension two. A key tool to prove it was
the so-called Berger inequality which asserts 
\begin{equation*}
\mathrm{area}(M,g)\geq \frac{2a^{2}}{\pi}\chi (M),
\end{equation*}
being $\chi (M)$ the Euler-Poincar\'{e} characteristic of $M$, for a $2-$
dimensional compact Riemannian manifold $(M,g)$ without conjugate points
before a fixed distance $a$ in the parameter of any unit geodesic, and the
equality holds only in case $(M,g)$ has constant sectional curvature $\frac{
\pi ^{2}}{a^{2}}$. This inequality was later generalized to higher
dimensions by Berger and independently by Green \cite[Proposition 5.64]
{Bes78} as follows 
\begin{equation}
\mathrm{Vol}(M,g)\geq \frac{a^{2}}{\pi ^{2}n(n-1)}\int_{M}S \, d\mu _{g},
\label{0.1}
\end{equation}
where $(M,g)$ is a compact Riemannian manifold of dimension $n$, scalar
curvature $S$, and without conjugate points before a fixed distance $a$ in
the parameter of any (unit) geodesic. Moreover, equality holds if and only
if $(M,g)$ has constant sectional curvature $\frac{\pi ^{2}}{a^{2}}$. Note
that Berger inequality is a direct consequence from (\ref{0.1}), via the
Gauss-Bonnet Theorem. The Berger-Green inequality can be written
equivalently as follows 
\begin{equation}
\mathrm{Vol}(UM,\hat{g})\geq \frac{a^{2}}{\pi ^{2}(n-1)}\int_{UM}\mathrm{
{\widetilde Ric}} \, d\mu _{\hat{g}},  \label{0.2}
\end{equation}
where $UM$ denotes the unit tangent bundle of $(M,g)$, $\hat{g}$ is the
restriction to $UM$ of the Sasaki metric from the tangent bundle $TM$, 
$\mathrm{{\widetilde Ric}}$ denotes the quadratic form associated to the Ricci tensor of 
$(M,g)$ and $d\mu _{\hat{g}}$ is the canonical measure associated with $\hat{g}$ (the
kinematic measure).

The Blaschke conjecture for spheres in any dimension remained open until
1978, when a nice proof was found using results of M. Berger, A. Weinstein
and C.T. Yang (see \cite{Kaz82} and references therein or \cite[Appendix D]
{Bes78}).

Inspired from this approach, in this paper we want to study conjugate points
on null geodesics on Lorentzian manifolds. Remember that no null geodesic in
any $2$-dimensional Lorentzian manifold has any conjugate point 
\cite[Lemma 10.45]{BeeEhrEas96}. On the other hand, it is not difficult to
see that there are no conjugate point on null geodesics in any Lorentzian
manifold of constant sectional curvature. For a general Lorentzian manifold,
conjugate points on null (or more generally nonspacelike) geodesics have
been sistematically studied \cite[Chapters 10 and 11]{BeeEhrEas96}, 
\cite[Chapter 10]{One83}, even from different approaches \cite{EhrKim94}. In
the compact Lorentzian case, as far as we know, not much is known about the
behaviour of conjugate points on null geodesics. We will introduce in this
paper a {\it new integral inequality} on a compact Lorentzian manifold
that admits a timelike conformal vector field which extends to the
Lorentzian case the Berger-Green inequality. We will apply this inequality
to get new information about the behaviour of conjugate points on null
geodesics in such compact Lorentzian manifolds.

Given an $n$-dimensional Lorentzian manifold $(M,g)$, $n\geq 3$, the null
congruence associated with a timelike vector field $K$, $C_{K}M$, is a
natural codimension two submanifold of the tangent bundle $TM$ (see section
2 for the definition and main properties). It was used in \cite{Har85} to
characterize Robertson-Walker space-times, and in \cite{Koc85} to study
infinitesimal null isotropy. The null congruence is an orientable manifold
(Proposition \ref{pro2.1}) and has the property that, for each null tangent
vector $v,$ there exists a unique $\lambda \in \R$ such that $\lambda
v\in C_{K}M,$ thus it can be seen as the manifold of all null tangent
directions of $(M,g).$

The null congruence can be endowed with a Lorentzian metric, $\hat{g}$,
namely the restriction to $C_{K}M$ of the Sasaki metric on $TM$ induced from
Lorentzian metric $g$. Several geometric properties of $(C_{K}M,\hat{g})$
are similar to the well known ones of the unit tangent bundle of a
Riemannian manifold; for instance, $C_{K}M$ is a fiber bundle with fiber a
sphere $\S ^{n-2}$ (note that $n$ is the dimension of $M$) and the
natural projection from $C_{K}M$ onto $M$ is a semi-Riemannian submersion in
the sense of Gray \cite{Gra67} and O'Neill \cite[Definition 7.44]{One83}.

It is well known that, in Riemannian Geometry, the unit tangent bundle is
invariant by the geodesic flow. In the Lorentzian case, $C_{K}M$ is not
invariant by the geodesic flow , in general, but it is if and only if the
vector field $K$ is assumed to be conformal (Proposition \ref{pro2.5}). This
fact is fundamental for our approach, because it permits to choose a
distinguished affine parameter for each null geodesic, which we will call 
{\it affine }$K\mathit{-}$\textit{parameter} (see discussion after
Proposition \ref{pro2.5}). Moreover, the existence of a timelike conformal
vector field permits us to prove (without assuming compactness) a Liouville
type theorem, i.e. the canonical measure associated to the Lorentzian
metric $\hat{g}$ on $C_{K}M$ is preserved by the geodesic flow (Theorem 
\ref{teo2.6}). This will be a technical key fact to get our main aims along
this paper. On the other hand, the study of conformal vector fields in
Lorentzian Geometry is a topic of special importance. Historically, it has
been developed mainly under assumptions of interest in Physics. From a
mathematical viewpoint, the particular case of Killing vector fields has
appeared as a useful tool to get classification theorems in some area of
Lorentzian Geometry \cite{Kam93} and \cite{San97}; moreover, standard
Lorentzian space forms (those with a time-orientable double semi-Riemannian
covering admitting a timelike Killing vector field) have been studied in 
\cite{KulRay85}. In the compact Lorentzian case, the assumption of the
existence of a timelike conformal vector field yields to geodesic
completeness \cite{RomSan95} (recall that a compact Lorentzian manifold may
be incomplete). Finally, note that the existence of a timelike conformal
vector field has been also used to classify compact Ricci-flat Lorentzian
manifolds \cite{RomSan96}, \cite{RomSan98}, and to study compact spacelike
hypersurfaces of constant mean curvature \cite{AliRomSan97}.

The main result of this paper is the following integral inequality (Theorem 
\ref{teo3.2})\vspace*{-1.1mm}
\begin{equation}
\mathrm{Vol}(C_{K}M,\hat{g})\geq \frac{a^{2}}{\pi ^{2}(n-2)}\int_{C_{K}M}\mathrm{{\widetilde Ric}} \, d\mu _{\hat{g}}  \label{0.3}
\end{equation}
where null geodesics of $(M,g)$ are assumed to have no conjugate point in $
[0,a)$ and \textrm{Ric} denotes the Ricci tensor of $(M,g)$.
Moreover, the equality can be characterized in terms of the null sectional
curvature. In Lorentzian geometry sectional curvature plays a quite
different role than in Riemanian geometry due to the fact that it cannot be
defined on null (i.e. degenerate) tangent planes. Harris introduced the
notion of null sectional curvature and pointed out its relationship with
usual sectional curvature \cite{Har82}. If a null congruence has been fixed,
then the null sectional curvature can be seen as a map on null tangent
planes (see comment after Remark \ref{re3}).

The remainder of this paper is devoted to show several applications of the
previous integral inequality (\ref{0.3}). In Theorem \ref{teo3.8} we give
another version of (\ref{0.3}) which permits us reprove the Berger-Green
inequality in the Riemannian case (Corollary \ref{cor3.11}). If the
conformal vector field $K$ is assumed to be Killing and null geodesics are
assumed to have no conjugate point in $[0,a)$, then (\ref{0.3}) may be
notably improved. Moreover, in this case, equality holds if and only if 
the universal Lorentzian covering of  
$(M,g)$ is globally isometric to a semi-Riemannian product $(\S
^{1}\times B,-g_{can}+g_{B})$ where $(B,g_{B})$ is $\S ^{n-1}(\frac{ah}{
\pi })$ where $h=1/\sqrt{-g(K,K)}$, (Corollary \ref{cor3.10}).

Moreover, we prove in Corollary \ref{cor3.12} that if a compact Lorentzian
manifold $(M,g)$, $\dim M\geq 3$, of constant scalar curvature $S_{0}$
admits a timelike Killing vector field $K$ and has no conjugate point on its
null geodesics then $S_{0}\leq 0$, and $S_{0}=0$ implies that the first
Betti number of $M$ is not zero and the Levi-Civita connection of $g$ is
Riemannian. So, we obtain a proper extension of Kamishima theorem 
\cite[Theorem A(1)]{Kam93} (see Remark \ref{re8} for more details).

As a practical application, we show some non trivial properties of the null
geodesics of a natural Lorentzian metric on the odd dimensional sphere $\S ^{2n+1}$; 
we prove the existence of a conjugate point along any null
geodesic. Furthermore, every conjugate point along any null geodesic occurs
at the same affine $K-$parameter, and we give an upper bound of it which is
independent of the point and the null direction. We show for the special
case $\S ^{3}$ that the null sectional curvature is $8$, and it is the
unique odd dimensional Lorentzian sphere with constant null sectional
curvature. Moreover on $\S ^{3}$, each one of its null geodesics, has no
conjugate point in $[0,\frac{\pi }{2\sqrt{2}})$ and the first conjugate
point is at $\frac{\pi }{2\sqrt{2}}$ in the affine $K-$parameter. Moreover the lightlike conjugate locus at any point of $\S^{2n+1}$ has been found to be an imbedded sphere $\S^{2n-1}$.

\section{NULL CONGRUENCE AND GEODESIC VECTOR FIELD}

Let $(M,g)$ be an $n(\geq2)-$dimensional Lorentzian manifold; that is a
(connected) smooth manifold $M$ endowed with a non-degenerate metric $g$
with signature ($-,+,.....,+).$ We shall write $\nabla $ for its Levi-Civita
connection and $R$ for its Riemannian curvature tensor (our convention on
the curvature tensor is $R(X,Y)Z=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla
_{X}Z-\nabla _{[X,Y]}Z$). 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 $K_{p}$ is
timelike for all $p\in M$.

Unless we indicate otherwise, $(M,g)$ will denote a Lorentzian manifold
which admits a timelike vector field $K.$ Recall \cite{Har85}, \cite{Koc85}
that the {\it null congruence associated to }$K$ is the set 
\begin{equation*}
C_{K}M=\left\{ v\in TM:g(v,v)=0\text{ and }g(v,K_{\pi v})=1\right\} ,
\end{equation*}
where $\pi :TM\longrightarrow M$ is the natural projection.

We put $(C_{K}M)_{p}=\left\{ v\in T_{p}M:g(v,v)=0\text{ and }
g(v,K_{p})=1\right\} $ for every $p\in M$.

In this section we develop several properties of the null congruence. The
first important result is Theorem \ref{teo2.6}, which is a Liouville type
theorem that allows us to handle properly some integrals on the null
congruence.

\begin{proposition}
\label{pro2.1}$C_{K}M$ is an orientable embedded submanifold of $TM$ with
dimension $2(n-1)$.
\end{proposition}

\begin{proof}
Let $F:TM\longrightarrow \R ^2$ be the map defined by $
F(v)=(g(v,v),g(v,K_{\pi v}))$. It is clear that $F$ is smooth and $C_{K}M$
is the inverse image of its regular value $(0,1)$. Being $TM$ an orientable
manifold, it suffices to construct two pointwise independent vector fields
along the embedding $C_{K}M\longrightarrow TM$, $\mathbf{A}$ and $\mathbf{K,}
$ such that 
\begin{equation*}
T_{v}TM=T_{v}C_{K}M\oplus W_{v}\text{ for all }v\in C_{K}M,
\end{equation*}
where $W_{v}=Span\left\{\mathbf{A}_v,\mathbf{K}_v \right\}$, 
\cite[Proposition 6.5.8]{AbrMarRat88}. In fact, $\mathbf{A}$ and $\mathbf{K}$
can be chosen as follows 
\begin{equation*}
\mathbf{A}_v=(v)_{v}\text{ and }\mathbf{K}_v=(K_{\pi v})_{v},
\end{equation*}
where $\left( \text{ }\right) _{v}:T_{\pi v}M\longrightarrow T_{v}T_{\pi v}M$
denotes the natural identification.
\end{proof}

\begin{proposition}
\label{pro2.2}$(C_{K}M,\pi ,M)$ is a fiber bundle with fibre type $\S
^{n-2}$ and structure group $O(n-1)$.
\end{proposition}

\begin{proof}
Let $E_{1},E_{2},...,E_{n-1},E_{n}=K$ be a local reference frame on an open
subset $U\subset M$ such that $g(E_{i},E_{j})=\delta _{ij}$ and $
g(E_{i},K)=0 $ for every $1\leq i,j\leq n-1$. We can define a local
trivialization as follows 
\begin{equation*}
\Psi :  \pi ^{-1}(U) \longrightarrow  U\times \S ^{n-2}, \; \; \; \;
v \mapsto \big(\pi v,1/\sqrt{-v_{n}}(v_{1},...,v_{n-1})\big),
\end{equation*}
where $v=\sum_{i=1}^{n}v_{i}E_{i}(\pi v)$. The result follows after choosing
a suitable family of such local trivializations \cite[Chapter 5]{Hus94}.
\end{proof}

Let $(M,g)$ be a semi-Riemannian manifold with Levi-Civita connection $
\nabla $. The {\it connector} $c$ associated with $\nabla $ is defined by 
\begin{equation*}
c:TTM  \longrightarrow  TM, \;\;\;\;\; X  \mapsto  \frac{\nabla \alpha }{dt}\mid _{0},
\end{equation*}
where $\alpha $ is a curve in $TM$ with $\alpha ^{\prime }(0)=X$ and $\frac{
\nabla \alpha }{dt}$ is the covariant derivative of the vector field $\alpha 
$ along the curve $\pi \circ \alpha $ on $M$. The map $c$ is well defined
and the pair $(c,\pi )$ is a vector bundle morphism from $\pi
_{TM}:TTM\longrightarrow TM$ to $\pi :TM\longrightarrow M$, being $\pi _{TM}$
the natural projection$.$ Moreover, if $(u)_{v}\in T_{v}TM$ where $u\in
T_{\pi v}M,$ then $c((u)_{v})=u$.

We consider the Sasaki metric $\hat{g}$ on $TM$ defined from $g$, 
\cite[Section 4.5]{YanIsh73} and \cite[Section 1.K]{Bes78} 
\begin{equation*}
\hat{g}(X,Y)=g(d\pi (X),d\pi (Y))+g(c(X),c(Y))\text{ \quad for all }
X,Y\in T_{v}TM.
\end{equation*}

Recall that $\pi :(TM,\hat{g})\longrightarrow (M,g)$ is a
semi-Riemannian submersion in the terminology of Gray \cite{Gra67} or
O'Neill \cite[Definition 7.44]{One83}. Let $c_{v}$ be the restriction $c\mid
_{T_{v}(TM)}$ for every $v\in TM$. We get the following $\hat{g}-$orthogonal
decomposition 
\begin{equation*}
T_{v}TM=\mathrm{Ker}\text{ }c_{v}\oplus T_{v}T_{\pi v}M.
\end{equation*}

If $(M,g)$ is a Riemannian manifold then $(TM,\hat{g})$ is also a
Riemannian manifold but if $(M,g)$ is Lorentzian then $(TM,\hat{g})$ is
a semi-Riemannian with index 2. We will see in the following result that the
restriction of $\hat{g}$ to $C_{K}M$ is a Lorentzian metric. In the
sequel, $\hat{g}$ also will represent the induced metric on $C_{K}M$
from the Sasaki one of $TM$.

\begin{proposition}
\label{pro2.3}$(C_{K}M,\hat{g})$ is a Lorentzian manifold and the
restriction of $\pi $ to $C_{K}M$ is a semi-Riemannian submersion with
spacelike fibers.
\end{proposition}

\begin{proof}
In the same notation as in proof of Proposition \ref{pro2.1}. We get $
\hat{g}(\mathbf{A}_v,\mathbf{A}_v)=0$, $\hat{g}(\mathbf{A}_v,
\mathbf{K}_v)=1$ and $\hat{g}(\mathbf{K}_v,\mathbf{K}_v)<0$. Therefore, at any $v\in C_{K}M$, 
$W_{v}$, is a Lorentzian
vector space with the induced metric from $\hat{g}$, and it is not
difficult to show that the direct sum decomposition 
\begin{equation}
T_{v}TM=T_{v}C_{K}M\oplus W_{v}  \label{63}
\end{equation}
is an orthogonal decomposition. This gives that $(C_{K}M,\hat{g})$ is a
Lorentzian manifold. 

Now, note that $T_{v}(C_{K}M)_{p}=\left\{ w\in T_{p}M:g(w,v)=g(w,K_{p})=0\right\}$, $p=\pi (v)$, 
which implies that the induced metric from $\hat{g}$ on $(C_{K}M)_{p}$ is Riemannian for all $p \in M$. 
On the other hand, if we denote by $\left[
T_{v}(C_{K}M)_{p}\right] ^{\perp }$ the orthogonal complement of $
T_{v}(C_{K}M)_{p}$ in $T_{v}C_{K}M$, then $\left[ T_{v}(C_{K}M)_{p}\right] ^{\perp }=\mathrm{Ker}c_{v}$, and therefore,  $\hat{g}(X,Y)=g(d\pi (X),d\pi (Y))$
holds for every $v\in C_{K}M$ and $X,Y\in \left[ T_{v}(C_{K}M)_{p}\right] ^{\perp }$.
\end{proof}

Next consider that $(M,g)$ is a compact Lorentzian manifold which admits a
timelike vector field $K$. Then, it is easily seen that $C_{K}M$ must be
also compact. As a nice application of previous result, if $\dim M\geq 3$, a
slight variation of \cite[Proposition A.III.5]{BerGauMaz71} permits us to
obtain the following Fubini type result 
\begin{equation}
\int_{C_{K}M}f \, d\mu _{\hat{g}}=\int_{M}\Big\{ \int_{\left( C_{K}M\right)
_{p}}f \, d\mu _{p}\Big\} \, d\mu _{g},  \label{0.4}
\end{equation}
where $f\in C^{0}(C_{K}M)$, $d\mu _{\hat{g}}$ is the canonical measure
induced from the Lorentzian metric $\hat{g}$ of $C_{K}M$ (equivalently,
the canonical measure induced from the $\hat{g}-$volume in the following
Proposition \ref{pro2.4}) and $d\mu _{p}$ (resp. $d\mu _{g}$) is the
canonical measure associated with $\hat{g}\mid _{\left( C_{K}M\right)
_{p}}$(resp. $g$)$.$ In particular, for every $f\in C^{0}(M)$ we obtain 
\begin{equation*}
\int_{C_{K}M}(f\circ \pi ) \, d\mu _{\hat{g}}=\int_{M}(f \, {\rm vol}_{K}) \, d\mu
_{g},
\end{equation*}
where ${\rm vol}_{K}:M\longrightarrow \R$ is defined by ${\rm vol}_{K}(p)=\mathrm{
Vol}\left( (C_{K}M)_{p},\hat{g}\right) $. It is not difficult to show
that $\left( (C_{K}M)_{p},\hat{g}\right) $ and $\left( \S ^{n-2},
\frac{1}{-g(K_{p},K_{p})}g_{can}\right) $ are isometric Riemannian manifolds
where $(\S ^{n-2},g_{can})$ is the standard Riemannian unit sphere, and
so 
\begin{equation}
{\rm vol}_{K}(p)=\beta (n-2)h^{n-2}(p),  \label{1}
\end{equation}
where $\beta (n-2)$ denotes $\mathrm{Vol}(\S ^{n-2},g_{can})$ and $
h:M\longrightarrow \R$ is given by 
\begin{equation}
h(p)=\frac{1}{\sqrt{-g(K_{p},K_{p})}}.  \label{3}
\end{equation}
Note that the Riemannian metrics on the fibers $(C_{K}M)_{p}$ depend on $
p\in M$. In particular, we have 
\begin{equation}
\mathrm{Vol}(C_{K}M,\hat{g})=\beta (n-2)\int_{M}h^{n-2} \, d\mu _{g}.
\label{2}
\end{equation}

Recall that the cotangent bundle $T^{*}M$, of an arbitrary manifold, carries
a natural symplectic structure given by $d\alpha $ where $\alpha $ is the
1-form $\alpha (Y)=-\pi _{T^{*}M}(Y)\left[ d\pi _{1}(Y)\right] $, being $\pi
_{T^{*}M}:TT^{*}M\longrightarrow T^{*}M$, $\pi _{1}:T^{*}M\longrightarrow M$
the natural projections and $Y\in TT^{*}M$.

If $(M,g)$ is a semi-Riemannian manifold, then $g$ determines a vector
bundle isomorphism $\flat$ from $TM$ onto $T^{*}M$ by putting $v \mapsto g(v,\ )$.
We call $\alpha _{g}$ the pull-back by $\flat $ of $\alpha $, then 
\begin{equation}
\alpha _{g}(X)=-g(v,d\pi (X)),  \label{4}
\end{equation}
where $X\in T_{v}(TM).$ Thus, $d\alpha _{g}$ is a symplectic form on $TM.$
The geodesic vector field, induced from the metric $g$ on $M,$ is the vector
field $Z_{g}\in \mathfrak{X}(TM)$ given by 
\begin{equation}
i_{Z_{g}}(d\alpha _{g})=dE,  \label{5}
\end{equation}
where $E(v)=\frac{1}{2}g(v,v)$. An important property of this vector field is 
\begin{equation}
d\pi (Z_{g})=\pi _{TM}(Z_{g}),  \label{6}
\end{equation}
where $\pi _{TM}:TTM\longrightarrow TM$ is the natural projection.

Finally, the flow $\left\{ \Phi _{t}\right\} $ of $Z_{g}$, given by $\Phi
_{t}(v)=\gamma _{v}^{\prime }(t)$, is called the {\it geodesic flow} of $
(M,g)$. If $(M,g)$ is assumed to be complete, then $Z_{g}$ is a complete
vector field.

\begin{proposition}
\label{pro2.4}Let $(M,g)$ be a Lorentzian manifold which admits a timelike
vector field $K$, then the $\hat{g}-$volume of $(C_{K}M,\hat{g})$ is
given, up to a constant multiple, by the $2(n-1)$ form 
\begin{equation*}
\beta _{g}\wedge \alpha _{g}\wedge (d\alpha _{g})^{n-2},
\end{equation*}
where $\beta _{g}(X)=-g(K_{\pi v},d\pi (X))$ for every $X\in T_{v}C_{K}M$
and $\alpha _{g}$ also denotes its restriction to $C_{K}M$.
\end{proposition}

\begin{proof}
It is well known that $\Omega =(d\alpha _{g})^{n}$ is, up to a constant
multiple, the $\hat{g}$-volume of $(TM,\hat{g})$. So $i_{\mathbf{K}
}i_{\mathbf{A}}\Omega $ is a volume form on $C_{K}M$, moreover it is the $
\hat{g}-$volume, up to a constant multiple. In fact, we take the
following vector fields along the inclusion $C_{K}M\hookrightarrow TM$
\begin{equation}
\mathbf{\xi }_{\mathbf{1}}=\frac{\mathbf{K}}{\sqrt{-\hat{g}(\mathbf{K,K)}
}} \;\; {\rm and}\;\; \mathbf{\xi }_{\mathbf{2}}=\sqrt{-\hat{g}(\mathbf{K,K)}}
\mathbf{A+\xi }_{\mathbf{1}}, \label{Xies}
\end{equation}
a straightforward computation gives that $\mathbf{\xi }_{\mathbf{1}}$ and $
\mathbf{\xi }_{\mathbf{2}}$ are $\hat{g}-$orthonormal with $\mathbf{\xi }
_{\mathbf{1}}$ timelike. Moreover $i_{\mathbf{K}}i_{\mathbf{A}}\Omega =i_{
\mathbf{\xi }_{\mathbf{1}}}i_{\mathbf{\xi }_{\mathbf{2}}}\Omega $ holds.
This shows that $i_{\mathbf{K}}i_{\mathbf{A}}\Omega $ is the $\hat{g}-$
volume, up to a constant multiple of $(C_{K}M,\hat{g})$.

Now, we get $\mathfrak{L}_{\mathbf{A}}\alpha _{g}=i_{\mathbf{A}}d\alpha _{g}+di_{
\mathbf{A}}\alpha _{g}=i_{\mathbf{A}}d\alpha _{g}$. On the other hand, a
direct calculation gives us $\mathfrak{L}_{\mathbf{A}}\alpha _{g}=\alpha _{g}$.
Therefore, 
\begin{equation*}
i_{\mathbf{A}}\Omega =n \alpha _{g}\wedge (d\alpha _{g})^{n-1}.
\end{equation*}

Thus $i_{\mathbf{K}}i_{\mathbf{A}}\Omega =n \left[ i_{\mathbf{K}}\alpha
_{g}\wedge (d\alpha _{g})^{n-1}-(n-1)\alpha _{g}\wedge i_{\mathbf{K}}d\alpha
_{g}\wedge (d\alpha _{g})^{n-2}\right] $. We end the proof taking into
account that $i_{\mathbf{K}}\alpha _{g}=0$ and $i_{\mathbf{K}}d\alpha _{g}=
\mathfrak{L}_{\mathbf{K}}\alpha _{g}=\beta _{g}$.
\end{proof}

Since $\hat{g}(Z_{g},\mathbf{A})=\hat{g}(Z_{g},\mathbf{K})=0$, we
obtain from (\ref{63}) that $Z_{g}(v)\in T_{v}C_{K}M$ for every $v\in C_{K}M$
and every timelike vector field $K$. Nevertheless, $C_{K}M$ may be not
invariant by any local geodesic flow. We will think about this fact in the
following result.

Recall that a conformal vector field on a semi-Riemannian manifold $(M,g)$
is a vector field $K$ such that $\mathfrak{L}_{K}g=\rho g,$ where $\rho $ is a
(necessarily smooth) function on $M$. Equivalently, this condition can be
written 
\begin{equation}
g(\nabla _{X}K,Y)+g(X,\nabla _{Y}K)=\rho g(X,Y),  \label{conf}
\end{equation}
for every $X,Y\in \mathfrak{X}(M)$.

\begin{proposition}
\label{pro2.5}Let $(M,g)$ be a Lorentzian manifold which admits a timelike
vector field $K$. Then, $C_{K}M$ is invariant by any local geodesic flow if
and only if $K$ is a conformal vector field.
\end{proposition}

\begin{proof}
If $K$ is a conformal vector field, then for every $v\in C_{K}M$, we get
from (\ref{conf}) that $\frac{d}{dt}g(\gamma _{v}^{^{\prime }},K_{\gamma
_{v}})=g(\gamma _{v}^{^{\prime }},\frac{\nabla K_{\gamma _{v}}}{dt})=\frac{1
}{2}(\rho \circ \gamma _{v})g(\gamma _{v}^{^{\prime }},\gamma _{v}^{^{\prime
}})=0,$ thus $g(\Phi _{t}(v),K_{\gamma _{v}(t)})=g(v,K_{\pi v})=1$.
Conversely, if $C_{K}M$ is invariant by any local geodesic flow, then an
easy computation, gives $(\mathfrak{L}_{K}g)(v,v)=0$ for every $v\in C_{K}M$.
Thus, from \cite[Theorem 1]{DajNom80}, there exists $\rho \in C^{\infty }(M)$
such that $\mathfrak{L}_{K}g=\rho g$.
\end{proof}

It is remarkable that for an arbitrary timelike vector field $K$, $C_{K}M$ is 
invariant by the differential of the flow of $K$ if and only if $K$ is Killing, 
althought we will not use it here.

Note that if $K$ is a timelike conformal vector field, then every null
geodesic $\gamma _{v}$ of $(M,g)$ with $\gamma _{v}^{\prime }(0)=v\in C_{K}M$
, gives rise to a null geodesic $\gamma _{v}^{\prime }$ of $(C_{K}M,\hat{
g})$. Furthermore, each null geodesic $\beta $ of $(M,g)$ may be
reparametrized to obtain a null geodesic $\alpha $ which satisfies $\alpha
^{\prime }(t)\in C_{K}M$ for all $t.$ In fact, from Proposition \ref{pro2.5}
, it can be shown that $g(\beta ^{\prime },K_{\beta })=a\in \R$, $a\neq
0.$ Thus, if we put $\alpha (t)=\beta (\frac{t}{a})$ we achieve $g(\alpha
^{\prime },K_{\alpha })=1.$ Thus, in the sequel, we always assume that a
null geodesic $\gamma $ of $(M,g)$ satisfies $g(\gamma ^{\prime },K_{\gamma
})=1$ and then we will call {\it affine} $K-${\it parameter} to the
parameter $t$ of $\gamma $.

Let $x:[0,a]\times (-\delta ,\delta )\rightarrow M$, $x(t,s)$, be a smooth
variation of a curve $\gamma $. We write $\partial _{t}x,$ $\partial _{s}x$
for $dx(\frac{\partial }{\partial t}),$ $dx(\frac{\partial }{\partial s})$
respectively, and denote covariant derivation of vector fields along $x$,
with respect to the directions $\frac{\partial }{\partial t},$ $\frac{
\partial }{\partial s}$ by $\nabla _{t},$ $\nabla _{s}$ respectively. It is
well known that $\nabla _{t}\partial _{s}x=\nabla _{s}\partial _{t}x$ (see
for instance, \cite[Proposition 4.44]{One83}).

In the sequel of this paper, the following Liouville type result will be a
key fact.

\begin{theorem}
\label{teo2.6}Let $(M,g)$ be a Lorentzian manifold which admits a timelike
conformal vector field $K$, then 
\begin{equation*}
\mathrm{div}(Z_{g}\mid _{C_{K}M})=0,
\end{equation*}
where $\mathrm{div}$ denotes the divergence operator on $(C_{K}M,\hat{g}
).$
\end{theorem}

\begin{proof}
In this proof $Z_{g}$ will be written for $Z_{g}\mid _{C_{K}M}$. From Proposition 
\ref{pro2.4} it suffices to show that 
\begin{equation*}
\mathfrak{L}_{Z_{g}}(\beta _{g}\wedge \alpha _{g}\wedge (d\alpha
_{g})^{n-2})=0.
\end{equation*}

From (\ref{4}), (\ref{5}) and (\ref{6}), taking into account that $\alpha_{g}(Z_{g})=-2E$ 
which vanishes on $C_{K}M$, the above condition is equivalent to 
\begin{equation}
\left( \mathfrak{L}_{Z_{g}}\beta _{g}\right) \wedge \alpha
_{g}\wedge (d\alpha _{g})^{n-2}=0.  \label{7}
\end{equation}

Let $v\in C_{K}M$ and $X\in T_{v}C_{K}M$. We take a curve $\lambda $ on $
C_{K}M$ such that $\lambda (0)=v$ and $\lambda ^{\prime }(0)=X.$ The following
formula gives a variation of $\gamma _{v}^{\prime }:[0,\varepsilon
]\longrightarrow C_{K}M$ for some $\varepsilon >0$ sufficiently small: 
\begin{equation*}
x:[0,\varepsilon ]\times (-\delta ,\delta ) \longrightarrow 
C_{K}M, 
\; \; (t,s) \mapsto \Phi _{t}(\lambda (s))=\gamma
_{\lambda (s)}^{\prime }(t).
\end{equation*}

Observe that all the longitudinal curves of $x$ are null geodesics of $
(C_{K}M,\hat{g}).$

One easily checks that $d\Phi _{\varepsilon }(X)=d\Phi _{\varepsilon
}(\lambda ^{\prime }(0))=\widetilde{J}(\varepsilon )$ where $\widetilde{J}$
is the variation vector field of $x$ (in fact, it is a Jacobi vector field
along $\gamma _{v}^{\prime }$). Now, in a natural way $x$ provides us with a
variation $\pi \circ x$ of $\gamma _{v}:[0,\varepsilon ]\longrightarrow M$
with null geodesics as longitudinal curves. Thus, we can get: 
\begin{equation*}
\big(\Phi _{\varepsilon }^{*}\beta _{g}\big)_{v}(X)=(\beta
_{g})_{\Phi _{\varepsilon }(v)}(d\Phi _{\varepsilon }(X))=-g(J(\varepsilon
),K_{\gamma _{v}(\varepsilon )}),
\end{equation*}
where $J$ is the variation vector field of $\pi \circ $ $x$ ($J$ is also a
Jacobi vector field on $\gamma _{v}$). Therefore 
\begin{equation}
\mathfrak{L}_{_{Z_{g}}}\beta _{g}(X)=-\frac{d}{dt}g(J,K_{\gamma
_{v}})\mid _{0}.  \label{8}
\end{equation}

Since  $\partial _{t}(\pi \circ x)(0,s)=\lambda (s)\in C_{K}M$, therefore
$g(\partial _{t}(\pi \circ x),K_{\pi \circ x})=1$, and thus 
\begin{equation*}
g\big( \nabla _{s}\partial _{t}(\pi \circ x)(t,s),K_{\pi \circ
x}(t,s)\big) +g\big( \partial _{t}(\pi \circ x)(t,s),\nabla _{s}K_{\pi
\circ x}(t,s)\big) =0.
\end{equation*}

If we take $t,s=0$ in previous equation, then 
\begin{equation}\label{nueva}
g\Big( K_{\pi v},\frac{\nabla J}{dt}\mid _{0}\Big) -g\Big( J(0),\frac{
\nabla K_{\gamma _{v}}}{dt}\mid _{0}\Big) +(\rho \circ \pi )(v)
g\left( v,J(0)\right) =0,
\end{equation}
where $\mathfrak{L}_{K}g=\rho \, g$. But $J(0)=d\pi (X)$ and  
\begin{equation*}
c(X)=\frac{\nabla \lambda }{ds}
\mid _{0}=\nabla _{s}\partial _{t}(\pi \circ x)(0,0)=\nabla _{t}\partial _{s}(\pi \circ x)(0,0)=\frac{\nabla J}{dt}\mid_{0}.
\end{equation*}
On the other hand, ${\hat g}(X,{\bf K}_v)=g(c(X),K_{\pi v})=0$. And thus, from
(\ref{nueva}), we obtain 

\begin{equation}
g\Big( J(0),\frac{\nabla K_{\gamma _{v}}}{dt}\mid _{0}\Big) =(\rho \circ
\pi )(v) g\left( v,J(0)\right).  \label{10}
\end{equation}

A direct computation using (\ref{8}) and (\ref{10}) gives $\mathfrak{L}_{_{Z_{g}}}\beta _{g}=(\rho \circ \pi )
\alpha _{g}$.  
\end{proof}


\begin{remark}
\label{re2}Observe that our proof in Theorem \ref{teo2.6} cannot be directly
deduced from the classical Liouville theorem in the Riemannian case 
\cite[Proposition 1.56]{Bes78}. In fact, the $\hat{g}-$volume of $
(C_{K}M,\hat{g})$ introduced in Proposition \ref{pro2.4}, clearly
depends on $K$ as it is shown by the presence of the 1-form $\beta _{g}$ in
its definition. Compare with \cite[Appendix]{AnDahHow96}.
\end{remark}

Assume that $(M,g)$ is a compact Lorentzian manifold and $K$ is a timelike
conformal vector field. In this case, $Z_{g}\mid _{C_{K}M}$ is a complete
vector field because $C_{K}M$ is compact. Therefore, as a consequence of
Theorem \ref{teo2.6}, we have 
\begin{equation}
\int_{C_{K}M}(f\circ \Phi _{t}) \, d\mu _{\hat{g}}=\int_{C_{K}M}f \, d\mu _{
\hat{g}} \label{Liouville}
\end{equation}
for every $f\in C^{0}(C_{K}M)$ and $t\in \R$.

\begin{remark} \label{re3}
In \cite{Koc85} the null congruence, associated with a
timelike vector field $K$, has been defined only when $g(K,K)=-1$, in this
case from (\ref{1}) and (\ref{2}), ${\rm vol}_{K}(p)=\beta (n-2)$ and $\mathrm{Vol}
(C_{K}M,\hat{g})=\beta (n-2)\mathrm{Vol}(M,g)$; nevertheless, if we take 
$U=hK$, then $g(U,U)=-1$ but $U$ may be no conformal, but as we have shown,
to be conformal is a crucial fact to get Proposition \ref{pro2.5} and
Theorem \ref{teo2.6} (see also \cite{Har85}).
\end{remark}

Now, we close this section by analyzing the well known notion of null
sectional curvature \cite[Definition A.6]{BeeEhrEas96}, \cite{Har85} in our
present setting to be used later. Let $(M,g)$ be a Lorentzian manifold, 
$\dim M\geq 3$. Given a null vector $v$ and a null plane $\sigma $ containing 
$v,$ 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 non-null (and therefore spacelike) vector in $\sigma$.
This is independent of the choice of the non-zero spacelike vector $u,$ but
it does depends quadratically on the choice of $v$ in $\sigma .$ However, if
a null congruence associated with a timelike vector field $K$ has been fixed
and we make the choice of $v\in C_{K}M\cap \sigma$, then the null sectional
curvature can be thought as a function on null planes. Along this paper we
always use null sectional curvature under that assumption and we will call
it the $K-${\it normalized null sectional curvature.}

It is well known that null sectional curvature is identically zero if and
only if the sectional curvature is constant \cite[Proposition 2.3]{Har82}.
On the other hand, a semi-Riemannian product $I\times B$ of a negative
definite open interval $(I,-dt^{2})$ and a Riemannian manifold $(B,g_{B})$
has constant $\frac{\partial }{\partial t}-$normalized null sectional
curvature $c$ if and only if $(B,g_{g})$ has constant sectional curvature $c$.


\section{THE INTEGRAL INEQUALITY}

In this section we prove a boundeness property for the volume of a compact
Lorentzian manifold which admits a timelike conformal vector field and with
no conjugate point along any null geodesic before a fixed length of its
affine $K-$parameter. As it is mentioned in the introduction, this will
resemble to a well known inequality of Berger-Green in the Riemannian case.
In fact, in the particular case that our Lorentzian manifold is the
product of $(\S ^1,-g_{can})$ and an arbitrary compact Riemannian
manifold $(B,g_{B})$, we obtain, as a consequence of our result, the
classical theorem of Berger-Green for $(B,g_{B})$.

We recall that, if $\gamma _{v}:[0,a]\longrightarrow M$ is a null geodesic
of $(M,g)$ such that there are no conjugate points of $\gamma _{v}(0)$ in $
[0,a)$, then the {\it Hessian form }$H_{\gamma _{v}}^{\perp }$ is
positive semidefinite, i.e. 
\begin{equation*}
H_{\gamma _{v}}^{\perp }(V,V)=\int_{0}^{a}\Big[ g\Big(\frac{\nabla V}{dt},\frac{
\nabla V}{dt}\Big)-g(R(V,\gamma _{v}^{\prime })\gamma _{v}^{\prime },V)\Big]
dt\geq 0,
\end{equation*}
for every piecewise smooth vector field $V$ along $\gamma _{v}$ such that $
V(0)=0$, $V(a)=0$ and $g(\gamma _{v}^{\prime },V)=0,$ \cite[p. 290-1]{One83}.

An standard argument permits us to show:

\begin{lemma}
\label{lem3.1}Let $(M,g)\,$be a Lorentzian manifold, and let $\gamma
_{v}:\left[ 0,a\right] \longrightarrow M$ be a null geodesic such that $
\gamma _{v}(0)$ has no conjugate point in $[0,a)$. Let $V$ be a piecewise
smooth vector field along $\gamma _{v}$ such that $V(0)=0$, $V(a)=0$, $
g(\gamma _{v}^{\prime },V)=0$, and $H_{\gamma _{v}}^{\perp }(V,V)=0$. Then $V
$ is smooth and $\frac{\nabla ^{2}V}{dt^{2}}+R(V,\gamma _{v}^{\prime
})\gamma _{v}^{\prime }=f\, \gamma _{v}^{\prime }$ where $f$ is a smooth
function.
\end{lemma}

Now, we are ready to prove the main result of this paper:

\begin{theorem}
\label{teo3.2}Let $(M,g)$ be an $n(\geq 3)-$dimensional compact Lorentzian manifold, 
which admits a timelike conformal vector field $K$. 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}
{\rm Vol}(C_{K}M,\hat{g})\geq \frac{a^{2}}{\pi ^{2}(n-2)}
\int_{C_{K}M}\mathrm{{\widetilde Ric}} \, d\mu _{\hat{g}}.  \label{13}
\end{equation}
Moreover, equality holds if and only if $(M,g)$ has $K-$normalized null
sectional curvature $\frac{\pi ^{2}}{a^{2}}$.
\end{theorem}

\begin{proof}
Let $\gamma _{v}:[0,a]\longrightarrow M$ be a null geodesic of $(M,g)$ such
that $v\in C_{K}M.$

We take a set of parallel vector fields $E_{3},...,E_{n}$ along $\gamma _{v}$
such that $g(E_{i},E_{j})=\delta _{ij}$ and $g(\gamma _{v}^{\prime
},E_{i})=0 $ for every $i,j$. Thus, we have [Proof of Lemma 8.9]{One83} 
\begin{equation*}
\mathrm{Ric}(\gamma _{v}^{\prime },\gamma _{v}^{\prime
})=\sum_{i=3}^{n}g(R(E_{i},\gamma _{v}^{\prime })\gamma _{v}^{\prime
},E_{i}).
\end{equation*}

If we put $X_{i}(t)=\sin \frac{\pi t}{a}E_{i}(t)$. An easy computation gives 
\begin{equation}
\sum_{i=3}^{n}H_{\gamma _{v}}^{\perp }(X_{i},X_{i})=(n-2)\frac{\pi ^{2}}{
a^{2}}\int_{0}^{a}\cos ^{2}\frac{\pi t}{a}dt-\int_{0}^{a}\sin ^{2}\frac{\pi t
}{a}\mathrm{Ric}(\gamma _{v}^{\prime },\gamma _{v}^{\prime })dt\geq 0.
\label{15}
\end{equation}

Now, we integrate previous inequality on the space $C_{K}M$ taking into account 
that $\gamma _{v}^{\prime }(t)=\Phi _{t}(v)$.
So, we obtain 
\begin{equation*}
(n-2)\frac{\pi ^{2}}{a^{2}}\int_{[0,a]\times C_{K}M}\cos ^{2}\frac{\pi t}{a}
dt \, d\mu _{\hat{g}}\geq \int_{[0,a]\times C_{K}M}\sin^{2}\frac{\pi t
}{a}\mathrm{Ric}(\Phi _{t}(v),\Phi _{t}(v))dt \, d\mu _{\hat{g}}.
\label{60}
\end{equation*}

From Fubini theorem and (\ref{Liouville}) we get that the above inequality is equivalent to 
\begin{equation*}
(n-2)\frac{\pi ^{2}}{a^{2}}\mathrm{Vol}(C_{K}M,\hat{g})\int_{0}^{a}\cos ^{2}\frac{\pi t
}{a}dt\geq \int_{C_{K}M}\mathrm{{\widetilde Ric}} \, d\mu _{\hat{g}}\int_{0}^{a}\sin^{2}\frac{
\pi t}{a}dt
\end{equation*}
If we notice that $\int_{0}^{a}\cos ^{2}\frac{\pi t}{a}dt=\int_{0}^{a}\sin^{2}
\frac{\pi t}{a}dt>0$, we obtain (\ref{13}).

We are left with the proof of the if and only if part in the equality case.
Note that equality holds in (\ref{13}) if and only if 
\begin{equation}
H_{\gamma _{v}}^{\perp }(X_{i},X_{i})=0\text{ for every} \ i.
\label{16}
\end{equation}

But, from Lemma \ref{lem3.1}, if (\ref{16}) holds, then we have $\frac{
\nabla ^{2}X_{i}}{dt^{2}}+R(X_{i},\gamma _{v}^{\prime })\gamma _{v}^{\prime
}=f_{i}\, \gamma _{v}^{\prime }$, where $f_{i}$ are smooth functions.
Consequently 
\begin{equation*}
g(R(X_{i},\gamma _{v}^{\prime })\gamma _{v}^{\prime },X_{i})(t)=\frac{\pi
^{2}}{a^{2}}\sin^{2}\frac{\pi t}{a},\;\text{ }t\in [0,a].
\end{equation*}

On the other hand, a direct computation gives 
\begin{equation*}
g(R(X_{i},\gamma _{v}^{\prime })\gamma _{v}^{\prime },X_{i})(t)=\sin ^{2}
\frac{\pi t}{a}g(R(E_{i},\gamma _{v}^{\prime })\gamma _{v}^{\prime
},E_{i})(t), t\in [0,a],
\end{equation*}
and hence $g(R(E_{i},\gamma _{v}^{\prime })\gamma _{v}^{\prime },E_{i})=\pi ^{2}/a^{2}$, for every $i=3,...,n$. Which implies that $\mathcal{K}_{v}(\sigma )=\pi ^{2}/a^{2}$ for every null plane with $v\in C_{K}M$. The converse is
trivial.
\end{proof}

\begin{remark}
\label{re5}We point out that if the equality holds in (\ref{13}) and put $
U=hK$, then $(M,g)$ 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 a point function. So, $(M,g)$ is {\it infinitesimally
null-isotropic relative to} $U$ in the terminology of L. Koch-Sen in \cite
{Koc85}. In that paper and in \cite{Har85}, it is shown that if $U$ is a
unit timelike vector field on a Lorentzian manifold $(M,g)$, $\dim M\geq 3,$
then $(M,g)$ is infinitesimally null-isotropic relative to $U$ if and only
if it is {\it infinitesimally isotropic relative to} $U$, which means 
\cite{Kar82} that 
\begin{equation}
R(X,Y)V=k\left\{ g(Y,V)X-g(X,V)Y\right\}   \label{16.1}
\end{equation}
and 
\begin{equation}
R(X,U)U=\theta X,  \label{16.2}
\end{equation}
for all $X,Y,V\in U^{\perp }$ where $k,\theta \in C^{\infty }(M)$. Moreover, 
$(M,g)$ is infinitesimally isotropic relative to $U$ if and only if all
planes containing $U$ have sectional curvature $-\theta $, and all planes
perpendicular to $U$ have sectional curvature $k$ and, in this case we get, 
\begin{equation}
\nu =\theta +k,  \label{17}
\end{equation}
where $\nu :M\longrightarrow \R$ is the function defined by $\mathcal{K}
_{v}(\sigma )$ with $v\in C_{U}M$ and $\sigma $ is a null plane such that $
v\in \sigma $, \cite{Koc85}. In \cite{Kar82} H. Karcher proved that, when $
\dim M\geq 4$, a Lorentzian manifold $(M,g)$ is infinitesimally isotropic
relative to $U$, with everywhere non zero null sectional curvature, if and
only if:

1. The distribution $U^{\perp }$ is integrable.

2. The integral manifolds of $U^{\perp }$ are totally umbilic and have
constant curvature.

3. $(M,g)$ is locally conformal to a flat Lorentzian space.

We would like to point out that the conclusion 1 does not remain true if it
is assumed $\dim M=3$ (see Remark \ref{re9} below).
\end{remark}

Next we are going to show several technical results which permit us to give
another interpretation for the right-hand side of integral formula (\ref{13}). 
Let $T$ be a symmetric bilinear form on $T_{p}M$. We write $\widetilde{T}(v)=T(v,v)$ 
for its associated quadratic form and $A_{T}$ for the linear operator 
given by $T(u,v)=g(A_{T}(u),v)$.

\begin{lemma}
\label{lem3.3}Let $(M,g)$ be an $n(\geq 3)-$dimensional Lorentzian
manifold admitting an arbitrary timelike vector field $K$ and $p\in M$.

(a) If $\varphi$ is a linear 1-form on $T_{p}M$, then
\begin{equation*}
\int_{(C_{K}M)_{p}}\varphi \, d\mu _{p}=-\beta (n-2)\varphi(K_{p})h(p)^n.
\end{equation*}

(b) If $T$ is a symmetric bilinear form  on $T_{p}M$, then 
\begin{equation*}
\int_{(C_{K}M)_{p}}\widetilde{T} \, d\mu _{p}=\beta (n-2)h(p)^n \Big[ \frac{
nh(p)^2\widetilde{T}(K_{p})+\mathrm{trace}\text{ }A_{T}}{(n-1)}\Big].
\end{equation*}
\end{lemma}

\begin{proof}
(a) To check this formula we are going to study the semi-Riemannian submanifold $
\left( (C_{K}M)_{p},g\right) $ of $\left( T_{p}M,g\right) $ (observe that $
\hat{g}\mid _{T_{p}M}=g)$. We take $\mathbf{\xi }_{\mathbf{1}}$ and $\mathbf{\xi }_{\mathbf{2}}$ as they were defined in (\ref{Xies}) but restricted to $(C_{K}M)_{p}$. 
Observe that they are $g-$orthonormal with $\mathbf{\xi }_{\mathbf{1}}$ timelike and 
\begin{equation*}
\left( T_{v}(C_{K}M)_{p}\right) ^{\perp }=Span\left\{ \mathbf{\xi }_{\mathbf{
1}}(v),\mathbf{\xi }_{\mathbf{2}}(v)\right\} \subset T_{v}T_{p}M.
\end{equation*}

The {\it Gauss formula} for the submanifold $(C_{K}M)_{p}$ in $T_{p}M$ is
written as follows 
\begin{equation}
\nabla _{X}^{0}Y=\nabla _{X}Y-h(p)^{-1}g(X,Y)\mathbf{\xi }_{
\mathbf{2}},  \label{18}
\end{equation}
where $X,Y\in \mathfrak{X}((C_{K}M)_{p})$ and $\nabla ^{0}$ (resp. $\nabla $) is
the induced connection on $(C_{K}M)_{p}\subset T_{p}M$ (resp. the
Levi-Civita connection of $g\mid _{(C_{K}M)_{p}}$).

Denote by ${\rm Hess}$ and ${\rm Hess}^{0}$ the Hessian operators of $
(C_{K}M)_{p}$ and $T_{p}M$ respectively. It is easily seen from (\ref{18}), taking into account ${\rm Hess}^0(\varphi)=0$, that 
\begin{equation}
{\rm Hess}\, \varphi(X,Y)=-h(p)^{-1}g(X,Y)
\mathbf{\xi }_{\mathbf{2}}(\varphi).  \label{19}
\end{equation}

A straightforward computation from (\ref{19}) gives for
the \textit{Laplacian} of $\varphi$
\begin{equation}
(\bigtriangleup \varphi)(v)=-(n-2)h(p)^{-1}{\mathbf{\xi }_{2}}_{v}(\varphi).  \label{20}
\end{equation}

Since $\varphi$ is lineal, it is easy to check that $\mathbf{A}(\varphi)=\varphi,$ $\mathbf{K}(\varphi)=\varphi(K_{p})$. Thus, from (\ref{20}), we get 
\begin{equation}
\bigtriangleup \varphi=(n-2)[g(K_{p},K_{p})\varphi-\varphi(K_{p})].  \label{22}
\end{equation}

We end the proof of (a) observing that our integral formula can directly achieved
from (\ref{22}) taking into account the clasical Green divergence theorem
and (\ref{1}).\\

(b) If $X,Y\in \mathfrak{X}((C_{K}M)_{p})$, then it is easily seen that
\begin{equation*}
Y(\widetilde{T})=2T(Y,\mathbf{A})\text{ and}
\end{equation*}
\begin{equation}
X(Y\widetilde{T})=2T(\nabla _{X}^{0}Y,\mathbf{A})+2T(X,Y).  \label{23}
\end{equation}

From the Gauss formula (\ref{18}) and (\ref{23}) we obtain
\begin{equation}
X(Y\widetilde{T})=2T(\nabla _{X}Y,\mathbf{A})-2h(p)^{-1}g(X,Y)T(\mathbf{A},
\mathbf{\xi }_{\mathbf{2}})+2T(X,Y).  \label{24}
\end{equation}

Therefore 
\begin{equation}
{\rm Hess}\widetilde{T}(X,Y)=-2h(p)^{-1}g(X,Y)T(\mathbf{A},\mathbf{\xi }_{\mathbf{
2}})+2T(X,Y).  \label{25}
\end{equation}

Let $\left\{ a_{1},...,a_{n-2}\right\} $ be an orthonormal basis of $
T_{v}(C_{K}M)_{p}$, then \linebreak $\left\{ a_{1},...,a_{n-2},\mathbf{\xi }
_{\mathbf{1}}(v),\mathbf{\xi }_{\mathbf{2}}(v)\right\} $ is also an
orthonormal basis of $T_{v}T_{p}M\equiv T_{p}M$ with $\mathbf{\xi }_{\mathbf{
1}}(v)$ timelike.
Now, by using (\ref{25})
\begin{equation}
(\bigtriangleup \widetilde{T})(v)=-2(n-2)h(p)^{-2}\widetilde{T}(v)-2(n-2)T(K_{p},v)+2
\sum_{i=1}^{n-2}\widetilde{T}(a_{i}).  \label{26}
\end{equation}

On the other hand 
\begin{equation*}
\mathrm{trace}\text{ }A_{T}=
\sum_{i=1}^{n-2}\widetilde{T}(a_{i})+h(p)^{-2}\widetilde{T}(v)+2T(v,K_{p}),
\end{equation*}
which permits to rewrite (\ref{26}) as follows 
\begin{equation*}
(\bigtriangleup \widetilde{T})(v)=-2(n-1)h{-2}(p)\widetilde{T}(v)-2nT(K_{p},v)+2\mathrm{trace}
A_{T},
\end{equation*}
and thus 
\begin{equation}
\int_{(C_{K}M)_{p}}\Big\{(1-n)h(p)^{-2}\widetilde{T} -nT(K_{p},\ )\Big\} d\mu _{p}+ 
\mathrm{trace}A_{T}\,{\rm vol}_{K}(p)=0 \label{27}
\end{equation}

Due to the fact that $T(K_{p},\ )$ is linear, (a) give us 
\begin{equation}
\int_{(C_{K}M)_{p}}T(K_{p},\ ) \, d\mu _{p}=-\widetilde{T}(K_{p})h(p)^n \beta (n-2).
\label{28}
\end{equation}

We end the proof substituting (\ref{28}) in (\ref{27}), taking into account (\ref{1}).
\end{proof}

Observe that if $\S ^k$ is a usual $k-$dimensional unit sphere of $\R ^{n+1}$, $1\leq k\leq n$, i.e. 
$\S ^k=\big\{ (x_{1},...,x_{n+1}):\sum_{i=1}^{k+1}x_{i}^{2}=1 \text{
and} \; x_{k+2}=...=x_{n+1}=0\big\}$ ,
and $\varphi$ is an 1-form on $\R ^{n+1}$, then clearly $\int_{\S ^k}\varphi \, d\mu =0$,
where $d\mu $ is the canonical measure of $\S ^k$. In particular, on
the contrary to Lemma \ref{lem3.3}, it holds 
$\int_{(UM)_{p}}\varphi \, d\mu _{p}=0$, for any Riemannian manifold $(M,g)$, where $p\in M$, $(UM)_{p}$ is the unit sphere in $T_{p}M$ and $\varphi$ is any 1-form on $T_{p}M$. Note that $(C_{K}M)_{p}$ is not centered at the origin of $T_{p}M$; in fact, $u\in (C_{K}M)_{p}$ implies $-u\notin (C_{K}M)_{p}$.

Remembering now the integral formula (\ref{0.4}) and using Lemma \ref{lem3.3}
for the Ricci tensor at any point $p\in M$, we get

\begin{equation}
\int_{C_{K}M}\mathrm{{\widetilde Ric}} \, d\mu _{\hat{g}}=\frac{\beta (n-2)}{n-1}
\int_{M}\left[ n\mathrm{Ric}(K,K)-g(K,K)S\right] h^{n+2} \, d\mu _{g}.
\label{29}
\end{equation}

Now, taking into account formula (\ref{29}) we are in a position to rewrite
Theorem \ref{teo3.2} as follows

\begin{theorem}
\label{teo3.8}Let $(M,g)$ be an $n(\geq 3)-$dimensional compact Lorentzian manifold,
which admits a timelike conformal vector field $K$. 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}\Big[ n
\mathrm{Ric}(K,K)-g(K,K)S\Big] h^{n+2} \, d\mu _{g}.  \label{30}
\end{equation}
Moreover, equality holds if and only if $(M,g)$ has $K-$normalized null
sectional curvature $\frac{\pi ^{2}}{a^{2}}$ 
\endproof
\end{theorem}

In a similar way, taking $T(u,v)=-g(R(u,U_{\pi v})U_{\pi v},v)$ where $U$
is an arbitrary unit timelike vector field, it is clear that for every $v\in C_{U}M$, $\widetilde{T}(v)$ is the sectional curvature of the non-degenerate plane $Span\left\{v,U_{\pi v}\right\}$. Thus, we obtain

\begin{corollary}
\label{cor3.7}Let $(M,g)$ be an $n(\geq 3)-$dimensional compact Lorentzian manifold,
which admits an unitary timelike vector field $U$. Then, 
\begin{equation*}
\int_{C_{U}M}\widetilde{T} \, d\mu _{\hat{g}}=-\frac{\beta (n-2)}{n-1}\int_{M}
\mathrm{Ric}(U,U) \, d\mu _{g}.
\end{equation*}
\endproof
\end{corollary}

In order to apply previous result to unit timelike vector fields of
geometric interest, note that if $K$ is a timelike conformal vector field
with $\mathfrak{L}_{K}g=\rho g$, then the vector field $U$ defined by $U=hK$ is
unitary and timelike. Moreover, it satisfies $(\mathfrak{L}_{U}g)(X,Y)=\rho
g(X,Y)$ for all $X,Y\in U^{\perp }$. Recall that a unit timelike vector
field $U$ is said to be a \textit{spatially conformal reference frame} \cite{GarKup96} 
if it satisfies that condition. Note that not
all spatially conformal reference frames can be obtained as
previously, \cite{RomSan98}.

By using Corollary \ref{cor3.7} we can rewrite \cite[Theorem 3.6]{RomSan98}
as follows:\medskip

\textit{If} $U$\textit{is a spatially conformal reference frame on an} $n(\geq3)-$\textit{dimensional compact Lorentzian manifold} $(M,g)$ \textit{then}

\begin{equation*}
\int_{C_{U}M}\widetilde{T}\, d\mu _{\hat{g}}\leq 0,
\end{equation*}
\textit{and equality holds if and only if }$\nabla _{v}U=0$ \textit{for
every }$v\in U^{\perp }$\textit{.\medskip}

On the other hand, from (\ref{29}) and Corollary \ref{cor3.7} we deduce
that the total scalar curvature of $(M,g)$ satifies 
\begin{equation*}
\int_{M}S \, d\mu _{g}=\frac{n-1}{\beta (n-2)}\int_{C_{U}M}\left[ \mathrm{{\widetilde Ric}}+n\widetilde{T} \right] \, d\mu _{\hat{g}}
\end{equation*}
for every unit timelike vector field $U$.

We have considered in all previous development timelike conformal vector
fields. Next, we focus our attention on the special case of timelike Killing
vector fields. Recall, a conformal vector field $K$ with $\mathfrak{L}_{K}g=\rho
g$ is said to be Killing if $\rho =0$.

\begin{lemma}
\label{lem3.9}Let $(M,g)$ be an $n-$dimensional compact Lorentzian manifold
which admits a timelike Killing vector field $K$ and put $U=hK$. We have 
\begin{equation}
\int_{M}h^{n}\mathrm{Ric}(U,U) \, d\mu _{g}\geq 0,  \label{33}
\end{equation}
and equality holds if and only if $U$ is parallel.
\end{lemma}

\begin{proof}
Since $\mathrm{div}\,U=0$, then the classical Raychaudhury equation 
\cite[p. 120]{SacWu77} for $U$ reduces to 
\begin{equation}
\mathrm{div}(\nabla _{U}U)=\mathrm{Ric}(U,U)+\mathrm{trace}\text{ }
(A_{U}^{\prime 2}),  \label{34}
\end{equation}
where $A_{U}^{\prime }$ is the linear operator of $U_{p}^{\bot }$ given by $
A_{U}^{\prime }(v)=-\nabla _{v}U$, for all $v\in U_{p}^{\bot }$.

From (\ref{34}) we can compute $\mathrm{div}(h^{n}\nabla _{U}U)$ and , after
using divergence theorem, we get 
\begin{equation}
\int_{M}h^{n}\mathrm{Ric}(U,U) \, d\mu _{g}=- \int_{M}\left( h^{n}\mathrm{
trace}\text{ }(A_{U}^{\prime 2})+(\nabla _{U}U)h^{n} \right) \, d\mu _{g}.  \label{35}
\end{equation}

On the other hand, an easy computation gives us that the adjoint operator of 
$A_{U}^{\prime }$ with respect to $g\mid _{U^{\perp }}$ is $-A_{U}^{\prime }$
and $\mathrm{grad}$ $h=-h^{3}\nabla _{K}K$. Therefore, if $e_{1}$,...,$
e_{n-1}$ is an orthonormal basis of $U_{p}^{\bot }$, then 
\begin{equation*}
\mathrm{trace}\left( A_{U}^{\prime 2}\right) =-\sum_{i=1}^{n-1}g\left(
A_{U}^{\prime }(e_{i}),A_{U}^{\prime }(e_{i})\right) \leq 0,
\end{equation*}
at any $p\in M$, and 
\begin{equation*}
(\nabla _{U}U)h^{n}=-nh^{n}g(\nabla _{U}U,\nabla _{U}U)\leq 0.
\end{equation*}

These inequalities imply the announced sign in (\ref{33}). Equality holds if
and only if $A_{U}^{\prime }=0$ and $\nabla _{U}U=0$.
\end{proof}

Note that if the equality holds in (\ref{33}) then $h$ must be constant.

\begin{corollary}
\label{cor3.10}Let $(M,g)$ be an $n(\geq 3)-$dimensional compact Lorentzian manifold which admits a timelike Killing vector field $K$. If 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 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{proof}
Since $K$ is a timelike Killing vector field we have, from (\ref{30}) and
Lemma \ref{lem3.9}, the inequality (\ref{36}). Moreover equality holds if and only 
if $U$ is parallel and $(M,g)$ has $K$-normalized null sectional curvature $\frac{\pi ^{2}}{a^{2}}$. In this case, if $(\widetilde{M},\widetilde{g})$ is the Lorentzian universal covering of $(M,g)$, then the result follows from the de Rham-Wu decomposition theorem \cite{Wu64}.
\end{proof}

Corollary \ref{cor3.10} contains, as a particular case, the classical
Berger-Green inequality in Riemannian geometry, (\cite{Gre63}, 
\cite[Proposition 5.64]{Bes78}).

\begin{corollary}
\label{cor3.11}Let $(B,g)$ be an $n(\geq 2)-$dimensional compact Riemannian manifold and scalar curvature $S$. Suppose no unit geodesic $\gamma : [0,a] \longrightarrow B$ 
has a conjugate point in $[0,a)$, then 
\begin{equation}
\text{\emph{Vol}}(B,g)\geq \frac{a^{2}}{\pi ^{2}n(n-1)}\int_{B}S \, d\mu _{g}.
\label{36.1}
\end{equation}
Moreover, equality holds if and only if $(B,g)$ has constant sectional
curvature $\frac{\pi ^{2}}{a^{2}}$.
\end{corollary}

\begin{proof}
This result easily follows if Corollary \ref{cor3.10} is specialized to the
Lorentzian manifold $(M,g_{L})=(\S ^1\times B,-g_{can}+g)$ where $
g_{can}$ is the canonical Riemannian metric on $\S ^1$, and taking $K$ the
lift to $\S ^1\times B$ of the vector field $z\mapsto iz$
on $\S ^1\subset \C$.
\end{proof}

\begin{corollary}
\label{cor3.12}Let $(M,g)$ be an $n(\geq 3)-$dimensional compact Lorentzian manifold
of constant scalar curvature $S=S_{0}$. If $(M,g)$ admits a timelike Killing
vector field $K$ and has no conjugate point on its null geodesics, then $
S_{0}\leq 0$, and $S_{0}=0$ implies that $U=hK$ is parallel, the first Betti
number of $M$ is not zero and the Levi-Civita connection of $g$ is
Riemannian.
\end{corollary}

\begin{proof}
Since there are no conjugate points on null geodesics, it follows from (\ref
{30}), 
\begin{equation}
\int_{M}\left[ n\mathrm{Ric}(U,U)+S_{0}\right] h^n \, d\mu _{g}\leq 0
\label{50}
\end{equation}
and taking into account (\ref{33}) we get $S_{0}\leq 0$. If $S_{0}=0$, then
it follows from (\ref{50}) and (\ref{33}), that $U=hK$ is parallel.
Therefore the Riemannian metric given by $g_{R}(X,Y)=g(X,Y)+2g(X,U)g(Y,U)$
for every $X,Y\in \mathfrak{X}(M)$ and the Lorentzian metric $g$ have the same
Levi-Civita connection. Let $\omega (X)=g(X,U)$ be the 1-form metrically
equivalent to $U$, $\omega $ is closed and if there is $f\in C^{\infty }(M)$
such that $\omega =df$, then $K_{p_{0}}=0$ at every critical point $p_{0}$
of $f$ and so $\omega $ is not exact.
\end{proof}

\begin{remark}
\label{re8}Kamishima proved \cite[Theorem A]{Kam93} that if a compact Lorentzian manifold $M$ with constant sectional curvature $c \in \R$ admits a timelike Killing vector field, then it is complete and $c \leq 0$. Moreover, if $c=0$ then $M$ is affinely diffeomorphic to a Riemannian manifold with non zero first Betti number. 
This result was widely extended in \cite[Corollaries 3.8, 3.9]{RomSan98}. A Lorentzian manifold of constant sectional curvature has no conjugate point along its null geodesics. Thus, Corollary \ref{cor3.12} is a proper extension, with a completely 
new technique, of Kamishima's theorem.
\end{remark}


\section{APPLICATION TO LORENTZIAN ODD DIMENSIONAL SPHERES}

In this section we give, as an application, some properties of null
geodesics of a natural Lorentzian metric on each odd dimensional sphere $
\S ^{2n+1}.$

We consider $\R ^{2n+2}$ identified to $\C ^{n+1}$ as usual: $
(x_{1},x_{2},...,x_{2n+1},x_{2n+2})=(z_{1},...,z_{n+1}),$ where $
z_{j}=x_{j}+ix_{n+1+j},$ $1\leq j\leq n+1,$ and we write 
$\S ^{2n+1}=\Big\{ z=(z_{1},...,z_{n+1})\in \C 
^{n+1}: \sum_{j=1}^{n+1}\left| z_{j}\right| ^{2}=1\Big\}$.

Let $U\in \mathfrak{X}(\S ^{2n+1})$ given by $U_{z}=iz$ at any $
z\in \S ^{2n+1}.$ For the canonical Riemannian metric $g_{can}$ of $\S ^{2n+1}$, $U$ is Killing and $g_{can}(U,U)=1.$ Therefore, $\nabla _{U}U=0,$
where $\nabla $ is the Levi-Civita connection of $g_{can},$ that is, the
integral curves of $U$ are geodesics for $g_{can}.$

We define on $\S ^{2n+1}$ a Lorentzian metric $g$ by 
\begin{equation}
g(X,Y)=g_{can}(X,Y)-2g_{can}(X,U)g_{can}(Y,U),  \label{38}
\end{equation}
for all $X,Y\in \mathfrak{X}(\S ^{2n+1})$. The vector field $U$ satisfies 
$g(U,U)=-1$, and it is easy to see that it is Killing for the Lorentzian
metric $g$. Thus $\widetilde{\nabla }_{U}U=0$, where $\widetilde{\nabla }$
is the Levi-Civita connection of $g$, and so, its integral curves are
timelike geodesics for $g$. Moreover, it is not difficult to show that 
\begin{equation}
\widetilde{\nabla}_{X}Y=\nabla _{X}Y-2g_{can}(X,U)
\nabla_{Y}U-2g_{can}(Y,U)\nabla _{X}U. \label{conexion}
\end{equation}

We consider the Hopf fibration 
\begin{equation*}
\pi :(\S ^{2n+1},g_{can})  \longrightarrow  (\C P^{n},g_{FS}), \; z \; \mapsto \; [z],
\end{equation*}
where $\C P^{n}$ is the complex projective space endowed with its classical
Fubini-Study metric $g_{FS}$ of constant holomorphic sectional curvature $4$ 
\cite[p. 273]{KobNom63}. The Hopf fibration is a principal bundle
with structural group $\S ^{1}$, and also a Riemannian submersion with totally geodesic fibers. If we change $g_{can}$ to the Lorentzian metric $g$, then $\pi$ 
is a semi-Riemannian submersion from $(\S ^{2n+1},g)$ to $(\C P^n,g_{FS})$ with timelike totally geodesics fibers.

It is useful to have an alternative definition for the metric $g$ in (\ref
{38}). In fact, we can also introduce $g$ from the Riemannina metric $g_{can}$ by putting 
\begin{equation*}
\begin{array}{l}
g_{\mid _{\mathcal{V}}}=-g_{can\mid _{\mathcal{V}}}, \;
g_{\mid _{\mathcal{H}}}=g_{can\mid _{\mathcal{H}}}, \;
g(\mathcal{V},\mathcal{H})=0,
\end{array}
\end{equation*}
where $\mathcal{V}$ is the vertical distribution and $\mathcal{H}$ is the
horizontal distribution for the canonical connection on the Hopf fibration.

Given a Lorentzian manifold $(M,g)$, a unit timelike vector field $U$ on $M$
and $p\in M$ recall that $(M,g)$ is said to be \textit{spatially isotropic }
with respect to $U$ at $p$ if for every two unit vectors $u_{1},u_{2}\in
U_{p}^{\bot }$ there is an isometry $\phi :M\longrightarrow M$ such that $
\phi (p)=p,$ $d\phi _{p}(U_{p})=U_{p}$ and $d\phi _{p}(u_{1})=u_{2}$. It is
said that $(M,g)$ is \textit{spatially isotropic} with respect to $U$ if it
is \textit{spatially isotropic} with respect to $U$ at every point, 
\cite[p. 47]{SacWu77}.

An easy algebraic argument gives

\begin{lemma}
\label{lem4.1}Let $(M,g)$ be a Lorentzian manifold which admits a unit
timelike vector field $U$. The following assertions are equivalent:

\begin{enumerate}
\item  $(M,g)$ is spatially isotropic with respect to $U$.

\item  For every $p\in M$ and for every $u,v\in (C_{U}M)_{p}$ there is an
isometry \newline $\left. \phi :M\longrightarrow M \right.$ such 
that $\phi (p)=p,$ $
d\phi _{p}(U_{p})=U_{p}$ and $d\phi _{p}(u)=v$. 
\endproof
\end{enumerate}
\end{lemma}

It is easy to show, using the previous result, that in the $3-$dimensional
case, $(M,g)$ spatially isotropic with respect to $U$ implies that it is
infinitesimally null-isotropic relative to $U$ (Remark \ref{re5}). But, as
Proposition \ref{pro4.2a} below shows, that assertion is not true in general
for bigger dimensions. In fact, according the conclusion (1) in Karcher
theorem (Remark \ref{re5}), if $(\S ^{2n+1},g)$ were infinitesimally
null-isotropic with respect to $U_{z}=iz$ and $2n+1>4$, then the
distribution $U^{\perp }$ must be integrable, which is not true. For the
sake of completeness, we will give a direct proof of the fact that $(\S
^{2n+1},g)$, with $2n+1>4$, is not infinitesimally null-isotropic with
respect to $U$, at the end of this section.

\begin{proposition}
\label{pro4.2a}$(\S ^{2n+1},g)$ is spatially isotropic with respect to $U
$ and the unitary group, $U(n+1),$ acts transitively by $g-$isometries on $
\S ^{2n+1}$.
\end{proposition}

\begin{proof}
It is well known that the unitary $U(n+1)$ acts transitively by $g_{can}-$
isometries on $\S ^{2n+1}$. Moreover, the vector field $U$ is $U(n+1)-$
invariant. Therefore, $U(n+1)$ also acts by $g-$isometries on $\S
^{2n+1} $. Thus, it is sufficient to check that $(\S ^{2n+1},g)$ is
spatially isotropic with respect to $U$ at the point $
e_{n+1}=(0,...,1)\in \S ^{2n+1}\subset \C ^{n+1}$.

Consider $U(n)=\left\{ A\in U(n+1):Ae_{n+1}=e_{n+1}\right\} $. If $u,v\in
U_{e_{n+1}}^{\perp }\subset \C ^{n+1}$, then $u=(u_{1},...,u_{n},0)$ and 
$v=(v_{1},...,v_{n},0)$. Hence, $u,v\in \S ^{2n-1}$ and so, there exists 
$A\in U(n)$ such that $Au=v$, giving the required isometry.
\end{proof}

We now find the null geodesics. Because of Proposition \ref{pro4.2a} it will be enough to find the null geodesics starting from a single point $p=(1,...,0)\in\S^{2n+1}\subset\C^{n+1}$. Firstly note that 
$v\in (C_{U}\S^{2n+1})_p$ if and only if $v=(-i,z_2 ,...,z_{n+1})$ with $\sum_{j=2}^{n+1} z_{j} \overline{z_j}=1$. On the other hand, (\ref{conexion}) yields
\begin{equation}
\nabla_{\gamma_{v}^{\prime}}\gamma_{v}^{\prime}=-4\nabla_{\gamma_{v}^{\prime}}U. \label{luz1} 
\end{equation}

The Gauss equation for the canonical Riemannian unit sphere permits us to write (\ref{luz1}) as follows
\begin{equation*}
\frac{d^2 \gamma_{v}}{dt^2}+4i\frac{d\gamma_v}{dt}-2\gamma_v =0.
\end{equation*}

We obtain for $\gamma_v =(\Theta_{1}^{v} ,...,\Theta_{n+1}^{v})$ with $\Theta_{j}^{v} :\R\rightarrow\C$, $(1\leq j\leq n+1)$
\begin{equation*}
\Theta_{1}^{v}(t)=\frac{2-\sqrt 2}{4}e^{(-2- \sqrt 2)it}+\frac{2+\sqrt 2}{4}e^{(-2+\sqrt 2)it},
\end{equation*}

\begin{equation*}
\Theta_{j}^{v}(t)=\frac{\sqrt{2}iz_j}{4}\Big[e^{(-2-\sqrt 2)it}-e^{(-2+ \sqrt 2)it}\Big]\ \ (j\geq2).
\end{equation*} 



\begin{corollary}
\label{cor4.3}There exists $\lambda \in (0,+\infty )$ such that for every
null geodesic $\gamma _{v}$ of $(\S ^{2n+1},g)$, $v\in C_{U}\S
^{2n+1}$, $\gamma _{v}(0)$ and $\gamma _{v}(\lambda )$ are conjugate and $
\gamma _{v}$ has no conjugate point on $[0,\lambda )$. Such $\lambda$ satisfies
\begin{equation}
\lambda ^{2}\leq \frac{(2n-1)\pi ^{2}}{4(n+1)},  \label{39}
\end{equation} 
and the past lightlike conjugate locus of all point $p\in\S^{2n+1}$ is an imbedded sphere $\S^{2n-1}$.
\end{corollary}

\begin{proof}
From \cite[Proposition 9.70]{Bes87}, the Ricci tensor and the scalar
curvature of $(\S ^{2n+1},g)$ satisfy 
\begin{equation}
\mathrm{Ric}(U,U)=2n,\text{ and }S=S_{FS}\circ \pi +2n,  \label{51}
\end{equation}
where $S_{FS}$ is the scalar curvature of $\C P^{n}$. But, we
know that $S_{FS}=4n(n+1)$ and hence the first assertion is a direct consequence of Corollary \ref {cor3.12} and Proposition \ref {pro4.2a}.
Now, the anounced inequality (\ref{39}) follows directly from Theorem \ref{teo3.8}. 

Finally, for any $u,v\in(C_{U}\S^{2n+1})_p$ with $u\not=v$, $\gamma_{u}(T)=\gamma_{v}(T)$ if and only if $T=\frac{q\pi}{\sqrt 2}$ for some $q\in \Z$, and so the last affirmation can be obtained using that $\lambda<\frac{\pi}{\sqrt 2}$.
\end{proof}

Note that, Corollary \ref{cor4.3} has a dual version for future null geodesics.
 
\begin{remark}
\label{re9}From Proposition \ref{pro4.2a} and (\ref{29}) we have that 
\begin{equation}
\mathrm{Ric}(v,v)=4(n+1)  \label{44}
\end{equation}
for every $v\in C_{U}\S ^{2n+1}$. Hence, (\ref{39}) also follows from 
\cite[Proposition 2.6]{Har82}.

For any $3-$dimensional Lorentzian manifold it is clear that 
\begin{equation}
\mathcal{K}_{v}(\sigma )=\mathrm{Ric}(v,v),  \label{45}
\end{equation}
for every null vector $v$ where $\sigma $ is the unique null tangent plane
containing $v$. Taking into account (\ref{44}) we conclude that the $U-$\textit{normalized null sectional curvature satisfies }$\mathcal{K}
_{v}(\sigma )=8$ \textit{for any degenerate plane} $\sigma $\textit{ of
the Lorentzian manifold} $(\S^3 ,g)$. Since the distribution $U^{\perp
}$ is not integrable, this shows that the hypothesis $\dim M\geq 4$ in
Karcher theorem \cite{Kar82} cannot be weakened to $\dim M\geq 3$.
\end{remark}

The following result is similar to the well-known Morse-Sch\"{o}nberg theorem 
\cite[Theorem 2.14]{Cha93}, and its proof follows from a slight variation to
the one in the Riemannian case. Compare with \cite[Proposition 2.6]{Har82}.

\begin{proposition}
\label{pro4.4}Let $(M,g)$ be a Lorentzian manifold, $\delta >0,$ and $\left.
\gamma :[0,\lambda ]\longrightarrow M\right. $ a null geodesic such that 
$\mathcal{K}_{\gamma ^{\prime }(t)}(\sigma )\leq \delta$,
for all null planes $\sigma $ with $\gamma ^{\prime }(t)\in \sigma $. If $
\gamma (0)$ and $\gamma (\lambda )$ are conjugate points along $\gamma $, then $\lambda \geq \frac{\pi }{\sqrt{\delta }}$.
\endproof
\end{proposition}

\begin{corollary}
\label{cor4.5}For every null geodesic $\gamma _{v}$ of $(\S ^3,g)$, $
v\in C_{U}\S ^3,$ $\gamma _{v}(0)$ and $\gamma _{v}(\frac{\pi }{2\sqrt{
2}})$ are conjugate points and $\gamma _{v}$ has no conjugate point on $[0,
\frac{\pi }{2\sqrt{2}})$. 
\endproof
\end{corollary}

Finally, note that $(\S ^3,g)$ is the unique Lorentzian odd
dimensional sphere which has constant $U-$normalized null sectional
curvature. This fact follows from Karcher's theorem in \cite{Kar82} and the
discussion before Proposition \ref{pro4.2a}. Nevertheless, we give here a direct proof of that statement. Any tangent plane containing $U$ admits a basis $\left\{ v,U_{\pi v}\right\} $ such that $v\in C_{U}\S ^{2n+1}$. On the other hand, from Proposition \ref{pro4.2a} $\theta (v)=-\mathcal{K}(v,U_{\pi v})$ is independent of $v$, and so it is a constant
function on $C_{U}\S ^{2n+1}$. Therefore, Corollary \ref{cor3.7} and (\ref{51}) give $\theta =1$.

If $X,Y\in U^{\perp }$ are orthogonal and unitary vector fields on $\S
^{2n+1}$, then the O'Neill formulas for a semi-Riemannian submersion, 
\cite[Theorem 7.47]{One83}, can be claimed to get 
\begin{equation}
\mathcal{K}\left( X,Y\right) =\mathcal{K}_{FS}\left( d\pi (X),d\pi (Y)\right) -\frac{3}{4}g\left( \mathcal{V}[X,Y],
\mathcal{V}[X,Y]\right) ,  \label{42}
\end{equation}
where we write $\mathcal{K}_{FS}$ for the sectional curvature of $(
\C P^n,g_{FS})$, and $\mathcal{V}$ denotes the vertical
operator. By a straightforward computation, we get 
\begin{equation*}
g(\mathcal{V}[X,Y],\mathcal{V}[X,Y])=-\frac{4}{3}\left[ \mathcal{K}_{FS}
\left( d\pi (X),d\pi (Y)\right) -1\right] .
\end{equation*}

Thus (\ref{42}) gives 
\begin{equation}
\mathcal{K}\left( X,Y\right) =2\mathcal{K}_{FS}\left( d\pi (X),d\pi (Y)\right) -1.  \label{43}
\end{equation}

From (\ref{43}), using the well-known range of variation of the sectional curvature of $\C P^{n}$, we get $1\leq \mathcal{K}(\tau )\leq 7$, for any tangent plane $\tau $ orthogonal to $U$. Taking into account (\ref{17}) we have 
\begin{equation}
2\leq \mathcal{K}_{v}(\sigma)\leq 8 \label{curvaturanula}
\end{equation}
for any degenerate plane $\sigma $ of the Lorentzian manifold $(\S ^{2n+1},g)$ with $v\in \sigma \cap C_{U}M$.

If $Span\left\{ d\pi (X),d\pi (Y)\right\} $ is a holomorphic tangent plane,
then $\mathcal{K}\left( X,Y\right) =7$. Hence, taking into account (\ref{17}), this fact gives,, another proof of $\mathcal{K}_{v}(\sigma )=8$ for any degenerate
plane $\sigma $ of $(\S ^{3},g)$ with $v\in \sigma \cap C_{U}M$.

If $n>1$ then $(\S^{2n+1},g)$ has not constant $U-$ normalized null sectional curvature. Thus, $(\S ^{2n+1},g)$ is not infinitesimally null-isotropic with respect to $U$. So, because of the characterization of the equality in Theorem \ref{teo3.8}, {\it inequality (\ref{39}) must be strict}. Using (\ref{curvaturanula}) and Proposition \ref{pro4.4} we get $\frac{\pi^2}{8}\leq\lambda^2 < \frac{(2n-1)\pi ^{2}}{4(n+1)}$.

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