text:
\section*{Introduction}
One calls a pseu\-\hbox{do\hs-}Riem\-ann\-i\-an manifold\ $\,(M,g)\,$ of dimension $\,n\ge4\,$ {\it conformally
symmetric\/} \cite{chaki-gupta} if its Weyl conformal tensor is parallel, and
{\it essentially conformally symmetric\/} if, in addition, $\,(M,g)\,$ is
neither conformally flat nor locally symmetric. All essentially con\-for\-mal\-ly symmetric\ manifold s have
indefinite metrics \cite[Theorem~2]{derdzinski-roter-77}.

The Weyl conformal tensor is one of the three irreducible components of the
curvature tensor under the action of the pseu\-do\hskip.7pt-or\-thog\-o\-nal group,
the other two corresponding to the scalar curvature and traceless Ricci
tensor. This puts con\-for\-mal\-ly symmetric\ manifold s on par with two other classes, formed by manifold s
with constant scalar curvature and, respectively, parallel Ric\-ci tensor,
including Einstein spaces.

The local structure of essentially con\-for\-mal\-ly symmetric\ pseu\-\hbox{do\hs-}Riem\-ann\-i\-an metric s is fully understood
\cite{derdzinski-roter-07}; they realize any prescribed indefinite signature
in every dimension $\,n\ge4$. They are also known to exist on some compact
manifolds dif\-feo\-mor\-phic\ to torus bundles over the circle \cite{derdzinski-roter},
where they represent all indefinite metric signatures in all dimensions
$\,n\ge5\,$ such that $\,\hbox{$n$\hskip3pt$\equiv$\hskip4pt$5$\hskip4.4pt$($mod\hskip2pt$3)$}$. Consequently, there arises a natural question
of characterizing the compact manifolds that admit such metrics.

The present paper provides a step toward an answer by establishing some
necessary conditions. Our first result, except for the claim about $\,\pi_1M$,
is derived in Section~\ref{pfta} from the Chern\hskip.7pt-Weil formulae.
Infiniteness of $\,\pi_1M\,$ is proved in Section~\ref{psta}: we argue there
that, if $\,M\,$ were simply connected, it would be a bundle over $\,S^2$ with
a fibre covered by $\,\text{\bf R}^{\hskip-.6ptn-2}\hskip-1pt$, which is impossible for
topological reasons.
\begin{theorem}\label{pontr}Let a manifold\/ $\hskip.7pt M\hskip-1pt$ of
\hskip-.3ptdimension $\hskip1ptn\ge4$ admit an essentially con\-for\-mal\-ly symmetric\ pseu\-\hbox{do\hs-}Riem\-ann\-i\-an metric. The real Pontryagin
classes $\,p_i(M)\in H^{4i}(M,\text{\bf R})$ then vanish for all\/ $\,i\ge1$, If, in
addition, $\,M\hskip.7pt$ is compact, then it has zero Euler characteristic, and its
fundamental group is infinite.
\end{theorem}
Applied to manifolds $\,M\,$ which are both compact and orientable, the
assertion about $\,p_i(M)\,$ in Theorem~\ref{pontr} leads to a similar
conclusion about the Pontryagin numbers, including the signature of $\,M$.

As mentioned above, compact essentially con\-for\-mal\-ly symmetric\ Lo\-rentz\-i\-an manifold s exist in infinitely
many dimensions $\,n$, starting from $\,n=5$, while noncompact ones exist in
every dimension $\,n\ge4$. Our next two results deal with their topological
structure in the compact case and with the dimension $\,4$.
\begin{theorem}\label{bdcir}Let\/ $\,(M,g)\,$ be a compact essentially con\-for\-mal\-ly symmetric\
Lo\-rentz\-i\-an manifold. Then some two-fold covering manifold of\/ $\,M\,$ is
the total space of a $\,C^\infty\hskip-1pt$ bundle over the circle, the fibre of
which admits a flat tor\-sion\-free connection with a nonzero parallel vector
field.
\end{theorem}
\begin{theorem}\label{nolor}Every four-di\-men\-sion\-al\ essentially con\-for\-mal\-ly symmetric\ Lo\-rentz\-i\-an manifold\ is
noncompact.
\end{theorem}
The fibration $\,M\to S^1$ in Theorem~\ref{bdcir} has an explicit geometric
description. Namely, its fibres are the leaves of a parallel distribution
$\,\dz^\perp$ on $\,(M,g)$, which is the orthogonal complement of the {\it Ol\-szak
distribution} $\,\mathcal{D}$, defined, in any con\-for\-mal\-ly symmetric\ manifold, by declaring the sections of
$\,\mathcal{D}\hskip.7pt$ to be the vector fields $\,u\,$ such that, for all vector fields
$\,v,v\hskip.4pt'\hskip-1pt$, one has $\,\xi\wedge\hskip.7pt\varOmega\hskip.7pt=0$, where
$\,\xi=g(u,\,\cdot\,)\,$ and
$\,\hskip.7pt\varOmega\hskip.7pt=W(v,v\hskip.4pt'\hskip-1pt,\,\cdot\,,\,\cdot\,)$. (Here $\,W\hskip.7pt$ denotes
the Weyl tensor; thus, $\,\hskip.7pt\varOmega\,$ is a differential $\,2$-form.)
Ol\-szak introduced the distribution $\,\mathcal{D}\,$ in a more general situation
\cite{olszak}, and showed that, on an essentially con\-for\-mal\-ly symmetric\ manifold, $\,\mathcal{D}\,$ is a null parallel distribution\ of
dimension $\,1\,$ or $\,2$. See Lemma~\ref{csnpd}(i).

We derive Theorems~\ref{bdcir} and~\ref{nolor} in Sections~\ref{potb}
and~\ref{potc} from the following result, proved in Section~\ref{pote}:
\begin{theorem}\label{parvf}For every essentially con\-for\-mal\-ly symmetric\ Lo\-rentz\-i\-an manifold, the Ol\-szak distribution\/
$\,\mathcal{D}$ is one-di\-men\-sion\-al. Passing to a two-fold covering manifold, if
necessary, we may assume that\/ $\,\mathcal{D}\hskip.7pt$ is trivial as a real line bundle,
and then\/ $\,\mathcal{D}\hskip.7pt$ is spanned by a global parallel vector field.
\end{theorem}

\renewcommand{\thetheorem}{\thesection.\arabic{theorem}}
\section{Preliminaries}\label{prel}
Throughout this paper, all manifolds and bundles, along with sections and
connections, are assumed to be of class $\,C^\infty\hskip-1pt$. Manifolds
(including fibres of bundles) are, by definition, connected. A mapping is
always a $\,C^\infty$ mapping betweeen manifolds.
\begin{remark}\label{fibra}A surjective submersion $\,\pi:M\to P\,$ such
that the sets $\,\pi^{-1}(y)$, $\,y\in P$, are all compact can always be
factored as $\,M\to Q\to P$, with a locally trivial fibration $\,M\to Q\,$
having compact (connected) fibres, and a finite covering projection
$\,Q\to P$.

Namely, $\,\pi\,$ itself is a locally trivial fibration, except that each
fibre $\,\pi^{-1}(y)$, rather than being connected (and hence a manifold in
our sense), may in general have some finite set $\,Q_y$ of connected
components. This well-known fact becomes clear if one uses the holonomy of
any ``nonlinear connection'' (a distribution in $\,M\,$ complementary to the
fibres). We now define $\,Q\,$ and $\,Q\to P\,$ by
$\,Q=\bigcup_{\hskip.4pt y\in P}(\{y\}\times Q_y)\,$ and $\,(y,N)\mapsto y$.
\end{remark}
\begin{lemma}\label{clext}Suppose that a closed\/ $\,1$-form\/ $\,\xi$ on
a compact manifold\/ $\,M\hskip.7pt$ is nonzero everywhere and, for some function
$\,\phi:M\to\text{\bf R}\,$ which is nonzero somewhere, the form\/ $\,\phi\hskip.7pt\xi\,$
is exact. Then\/ $\,M\hskip.7pt$ is the total space of a bundle over the circle
$\,S^1\hskip-1pt$. In addition, for any functions\/ $\,t:\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\to\text{\bf R}\,$ on the
universal covering manifold\ of\/ $\,M\hskip.7pt$ and\/ $\,\theta:M\to\text{\bf R}\hskip.7pt$, such
that\/ $\,dt\,$ is the pull\-back of\/ $\,\xi\,$ to $\,\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\,$ and\/
$\,d\theta=\phi\hskip.7pt\xi$, and for some $\,c\in(0,\infty)$,
\begin{enumerate}
  \def{\rm\alph{enumi}}{{\rm\alph{enumi}}}
\item $t:\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\to\text{\bf R}\,$ descends to a bundle projection
$\,M\to\text{\bf R}/c\hskip.4pt\text{\bf Z}\hskip.4pt=S^1\hskip-1pt$,
\item the pull\-back of\/ $\,\theta\,$ to\/ $\,\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\hskip.7pt$ equals the composite
$\,\varLambda(t)\,$ for some nonconstant function $\,\varLambda:\text{\bf R}\to\text{\bf R}\,$ which is
periodic, and has\/ $\,c\,$ as a period,
\item $\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\,$ can be dif\-feo\-mor\-phi\-cal\-ly\ identified with $\,\text{\bf R}\times N\hskip.7pt$ for some
manifold\/ $\,N\hskip.7pt$ so as to make\/ $\,\hskip.7pt t\,$ coincide with the
projection\/ $\,\text{\bf R}\times N\to\text{\bf R}\hskip.7pt$.
\end{enumerate}
\end{lemma}
\begin{proof}Let $\,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\,$ be the pull\-back to $\,\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\,$ of any fixed
Riemannian metric $\,g\,$ on $\,M$. The $\,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt$-gra\-di\-ent $\,\hskip.7pt\widehat{\hskip-.7pt\nabla\hskip-.4pt}\hskip.4pt t\,$ of
$\,t\,$ gives rise to the vector field $\,w=\hskip.7pt\widehat{\hskip-.7pt\nabla\hskip-.4pt}\hskip.4pt t/\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt(\hskip.7pt\widehat{\hskip-.7pt\nabla\hskip-.4pt}\hskip.4pt t,\hskip.7pt\widehat{\hskip-.7pt\nabla\hskip-.4pt}\hskip.4pt t)$, which is
complete, being the pull\-back of the vector field $\,u/g(u,u)\,$ on $\,M$,
for $\,u\,$ such that $\,\xi=g(u,\,\cdot\,)$. A standard argument
\cite[p.\ 12]{milnor} using the flow of $\,w$, for which $\,t\,$ itself serves
as the parameter, yields (c). Denoting by $\,\widehat\theta\,$ and
$\,\widehat\phi\,$ the pull\-backs of $\,\theta\hskip.7pt$ and $\,\phi\hskip.7pt$ to $\,\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt$,
we have $\,\hskip.7pt\widehat{\hskip-.7pt\nabla\hskip-.4pt}\hskip.4pt\widehat\theta=\widehat\phi\hskip.7pt\hskip.7pt\widehat{\hskip-.7pt\nabla\hskip-.4pt}\hskip.4pt t$. Hence $\,\widehat\theta\,$
is, locally, a function of $\,t$. The word `locally' can
in turn be dropped as the level sets of $\,t\,$ are connected by (c). Thus,
$\,\widehat\theta=\varLambda(t)\,$ (that is, $\,\widehat\theta=\varLambda\hskip-1pt\circ t$) for
some function $\,\varLambda:\text{\bf R}\to\text{\bf R}\hskip.7pt$. Since $\,\,\theta\,$ and
$\,\,\widehat\theta\,$ are nonconstant, so is $\,\varLambda$.

The invariance of $\,dt\,$ under the action of the deck transformation group
$\,\Gamma=\pi_1M$ implies that $\,t\circ\alpha=t+\varXi(\alpha)\,$ for some
homomorphism $\,\varXi:\Gamma\to\text{\bf R}\,$ and all $\,\alpha\in\Gamma$. As
$\,\widehat\theta=\varLambda(t)\,$ is $\,\Gamma$-in\-var\-i\-ant, every nonzero value of
$\,\varXi\,$ is a period of $\,\varLambda$. Thus, $\,\varXi(\Gamma)=c\hskip.4pt\text{\bf Z}$ for
some $\,c\in(0,\infty)$, and (b) follows. Finally, the surjective submersion
$\,t:\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\to\text{\bf R}$ descends to a mapping $\,M=\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\hskip-1pt/\Gamma\to\text{\bf R}/c\hskip.4pt\text{\bf Z}$,
which must be a surjective submersion as well. In addition, for each
$\,s\in\text{\bf R}\hskip.7pt$, the pre\-im\-age of $\,s+c\hskip.4pt\text{\bf Z}\,$ under the latter mapping
is connected, as it coincides with the image of $\,t^{-1}(s)\subset\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\,$
under the covering projection $\,\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\to M$, and $\,t^{-1}(s)\subset\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\,$
is connected by (c). Combined with Remark~\ref{fibra}, this proves (a).
\end{proof}
Given a connection $\,\nabla\,$ in a vector bundle\ $\,\mathcal{E}\hskip.7pt$ over a manifold\ $\,M$,
a section $\,\psi\,$ of $\,\mathcal{E}$, and vector field s $\,u,v\,$ tangent to $\,M$,
our sign convention for the curvature tensor $\,R=R^\nabla$ is
\begin{equation}\label{cur}
R(u,v)\psi\hskip7pt=\hskip7pt\nabla_{\!v}\nabla_{\!u}\psi\,
-\,\nabla_{\!u}\nabla_{\!v}\psi\,+\,\nabla_{[u,v]}\psi\hskip.7pt.
\end{equation}
Such $\,\nabla\hskip-.7pt,\hskip.7pt\mathcal{E},M,u\,$ and $\,v\,$ give rise to the bundle morphism
\begin{equation}\label{cvo}
R^\nabla\hskip-1pt(u,v):\mathcal{E}\,\to\,\mathcal{E}
\end{equation}
sending a section $\,\psi\,$ of $\,\mathcal{E}\hskip.7pt$ to
$\,R^\nabla\hskip-1pt(u,v)\psi=R(u,v)\psi\,$ defined
in (\ref{cur}).

We always denote by $\,\nabla\,$ both the Le\-vi-Ci\-vi\-ta connection\ of a given pseu\-\hbox{do\hs-}Riem\-ann\-i\-an manifold\ $\,(M,g)$,
and the $\,g$-gra\-di\-ent operator. The same symbol $\,\nabla\,$ is also
used for connections induced by $\,\nabla\,$ in $\,\nabla$-par\-al\-lel\ subbundles of
$\,{T\hskip-.3ptM}\,$ and their quotients.

A {\it pseu\-\hbox{do\hskip.7pt-}Riem\-ann\-i\-an fibre metric\/} $\,\gamma\,$ in a vector bundle\
$\,\mathcal{E}\,$ over a manifold\ $\,M\,$ is, as usual, any family of
nondegenerate symmetric bilinear forms $\,\gamma_x$ in the fibres
$\,\mathcal{E}_x$ that constitutes a $\,C^\infty$ section of the symmetric
power $\,(\mathcal{E}^*){}^{\odot2}\hskip-1pt$.
\begin{remark}\label{scfco}Every simply connected manifold $\,N$ with a
complete flat tor\-sion\-free connection $\,\nabla\,$ is dif\-feo\-mor\-phic\ to a
Euclidean space: the exponential mapping of $\,\nabla\hskip.7pt$ at any point
$\,x\in N\hskip.7pt$ is an affine dif\-feo\-mor\-phism\ $\,{T\hskip-2pt_x\hskip-.9ptN}\to N\hskip-.7pt$. See
\cite[p.\ 145]{auslander-markus}.
\end{remark}
Given a flat connection $\,\nabla\,$ in a vector bundle\ $\,\mathcal{E}\hskip.7pt$ of fibre dimension
$\,k\,$ over a manifold\ $\,M$, we will say that $\,\mathcal{E}\hskip.7pt$ {\it is trivialized by
its parallel sections\/} if the space of $\,\nabla$-par\-al\-lel\ global sections of
$\,\mathcal{E}\hskip.7pt$ is $\,k$-di\-men\-sion\-al\ (or, in other words, $\,\nabla\,$ is {\it globally}
flat).
\begin{lemma}\label{flcpl}If a compact manifold\/ $\,N$ with a flat
tor\-sion\-free connection\/ $\,\nabla\hskip.7pt$ admits a $\,\nabla$-par\-al\-lel\
distribution $\,\mathcal{L}\hskip.7pt$ such that both bundles $\,\mathcal{L}\hskip.7pt$ and\/
$\,\mathcal{E}={T\hskip-.3ptN}\hskip-.7pt/\mathcal{L}$, with the flat connections induced by $\,\nabla$, are
trivialized by their parallel sections, then\/ $\,\nabla\hskip.7pt$ is complete.
\end{lemma}
\begin{proof}We denote by $\,\mathcal{X}\,$ the space of parallel sections of
$\,\mathcal{L}$, and by $\,\hskip.7pt\exp_{\hskip.7pt x}$ the exponential mapping of $\,\nabla\hskip.7pt$
at $\,x\in M$. Geodesics tangent to $\,\mathcal{L}\,$ at some (or every) point, being
integral curves of elements of $\,\mathcal{X}$, are obviously complete.

Let $\,\mathcal{Y}\,$ be a vector subbundle of $\,{T\hskip-.3ptN}\hskip.7pt$ such that
$\,{T\hskip-.3ptN}=\mathcal{L}\oplus\mathcal{Y}$, and let us fix $\,w\inV\hskip-.7pt$, where $\,V\hskip.7pt$ is
the vector space of sections of $\,\mathcal{Y}\,$ obtained as the image of the
space of parallel sections of $\,\mathcal{E}\hskip.7pt$ under the obvious isomorphism
$\,\mathcal{E}\to\mathcal{Y}$. The vector field $\,v=\nabla_{\!w}w\,$ then is a section of
$\,\mathcal{L}$. (In fact, locally, $\,w=w\hskip.4pt'\hskip-.7pt+\widetilde{\hskip-.7pt w\hskip-.7pt}$, for a
section $\,w\hskip.4pt'$ of $\,\mathcal{L}\,$ and a local parallel vector field\
$\,\widetilde{\hskip-.7pt w\hskip-.7pt}$, so that $\,\nabla_{\!w}w=\nabla_{\!w}w\hskip.4pt'\hskip-1pt$,
while $\,\nabla_{\!w}w\hskip.4pt'$ is a section of $\,\mathcal{L}$.) Any integral curve
$\,\text{\bf R}\ni t\mapsto x(t)\in N\hskip.7pt$ of $\,w\,$ now gives rise to a function
$\,\zeta:\text{\bf R}\to\mathcal{X}\,$ defined by requiring that $\,\zeta(t)\in\mathcal{X}\,$ have the
value $\,v_{x(t)}$ at the point $\,x(t)$. Any function $\,\eta:\text{\bf R}\to\mathcal{X}$
with the second derivative $\,\ddot\eta=-\hskip.7pt\zeta\,$ leads in turn to the
curve $\,\text{\bf R}\ni t\mapsto y(t)\in N\hskip-.7pt$, given by
$\,y(t)=\exp_{\hskip.7pt x(t)}\hskip.4pt\eta(t,x(t))$, where $\,\eta(t,x)\in {T\hskip-2pt_x\hskip-.9ptM}\,$ is
the value of $\,\eta(t)\in\mathcal{X}\,$ at $\,x\in M$. That $\,t\mapsto y(t)\,$ is a
geodesic is clear: a treating $\,N\hskip-.7pt$, locally, as an affine space, we have
$\,y(t)=x(t)+\eta(t)$, and $\,\ddot y=\ddot x+\ddot\eta=0$. Since such
geodesics realize all initial data, our assertion follows.
\end{proof}

Let $\,(t,s)\mapsto x(s,t)\,$ be a fixed {\it variation of curves} in a pseu\-\hbox{do\hs-}Riem\-ann\-i\-an manifold\
$\,(M,g)$, that is, an $\,M$-val\-ued $\,C^\infty$ mapping from a rectangle
(product of intervals) in the $\,ts\hskip.4pt$-plane. By a {\it vector field\/ $\,w\,$
along the variation} we mean, as usual, a section of the pull\-back of
$\,{T\hskip-.3ptM}\,$ to the rectangle (so that $\,w(t,s)\in{T\hskip-2pt_{x(t,s)}\hskip-.9ptM}$). Examples are $\,x_s$
and $\,x_t$, which assign to $\,(t,s)\,$ the velocity of the curve
$\,t\mapsto x(t,s)\,$ (or, $\,s\mapsto x(t,s)$) at $\,s\,$ (or $\,t$). Further
examples are provided by restrictions to the variation of vector fields on
$\,M$. The partial covariant derivatives of a vector field $\,w\,$ along the
variation are the vector fields $\,w_t,\hskip.7pt w_s$ along the variation, obtained
by differentiating $\,w\,$ covariantly along the curves $\,t\mapsto x(t,s)\,$
or $\,s\mapsto x(t,s)$. Skipping parentheses, we write $\,w_{ts},\hskip.7pt w_{stt}$,
etc., rather than $\,(w_t){}_s,\hskip.7pt ((w_s){}_t){}_t$ for higher-or\-der
derivatives, as well as $\,x_{ss},\hskip.7pt x_{st}$ instead of
$\,(x_s){}_s,\hskip.7pt (x_s){}_t$. One always has $\,w_{ts}=w_{st}+R(x_t,x_s)\hskip.4pt w$,
cf.\ \cite[formula (11.2) on p.\ 493]{dillen-verstraelen}, and, since the
Le\-vi-Ci\-vi\-ta connection\ $\,\nabla\,$ is tor\-sion\-free, $\,x_{st}=x_{ts}$. Consequently,
$\,x_{ttss}\,=\,x_{stts}\,+\,[R(x_t,x_s)x_t]_s$,
$\,x_{stts}\,=\,x_{stst}\,+\,R(x_t,x_s)x_{st}$ and
$\,x_{stst}\,=\,x_{sstt}\,+\,[R(x_t,x_s)x_s]_t$. Thus, whenever
$\,(t,s)\mapsto x(s,t)\,$ is a variation of curves in $\,M$,
\begin{equation}\label{xts}
\begin{array}{rl}
\mathrm{a)}&\hskip10ptx_{tts}\,\,=\,\,x_{stt}\,+\,\hskip.7pt R(x_t,x_s)x_t\hskip.7pt,\\
\mathrm{b)}&\hskip10ptx_{ttss}\,\,=\,\,x_{sstt}\,+\,\hskip.7pt[R(x_t,x_s)x_s]_t
+\,\hskip.7pt R(x_t,x_s)x_{st}\,+\,\hskip.7pt[R(x_t,x_s)x_t]_s\hskip.7pt.
\end{array}
\end{equation}

\section{Con\-for\-mal\-ly symmetric manifolds}\label{csma}
The Schouten tensor $\,\sigma\,$ and Weyl conformal tensor $\,W\hskip.7pt$ of a pseu\-\hbox{do\hs-}Riem\-ann\-i\-an manifold\
$\,(M,g)\,$ of dimension $\,n\ge4\,$ are given by
$\,\sigma={\rho}\,-\hskip.7pt(2n-2)^{-1}\,\text{\rm s}\hskip1.2ptg$, with $\,{\rho}\,$
denoting the Ric\-ci tensor,
$\,\hskip.7pt\text{\rm s}\hskip.7pt=\hskip.7pt\text{\rm tr}_g\hskip.4pt{\rho}\hskip.7pt\,$ standing for the
scalar curvature, and $\,W\hskip-.7pt=R-(n-2)^{-1}\hskip.7pt g\wedge\hskip.4pt\sigma$. Here
$\,\wedge\,$ is the exterior multiplication of $\,1$-forms valued in
$\,1$-forms, which involves the ordinary $\,\wedge\,$ as the val\-ue\-wise
multiplication; thus, $\,g\wedge\hskip.4pt\sigma\,$ is a $\,2$-form valued in
$\,2$-forms.
\begin{lemma}\label{rwcod}For any essentially con\-for\-mal\-ly symmetric\ manifold of dimension $\,n\ge4$,
\begin{enumerate}
  \def{\rm\alph{enumi}}{{\rm\alph{enumi}}}
\item $R\hskip.7pt\,=\,\hskip.7pt W\hskip.7pt+\,(n-2)^{-1}\hskip.7pt g\wedge{\rho}$,
\item the Ric\-ci tensor $\,{\rho}\,$ satisfies the Co\-daz\-zi equation, in the
sense that the three-times covariant tensor field\/ $\,\nabla\hskip-.7pt{\rho}\,$ is
totally symmetric.
\end{enumerate}
\end{lemma}
\begin{proof}In any essentially con\-for\-mal\-ly symmetric\ manifold, $\,\hskip.7pt\text{\rm s}\hskip.7pt=0\,$ identically
\cite[Theorem~7, p.\ 21]{derdzinski-roter-78}, so that $\,\sigma={\rho}$. This
gives (a), and (b) follows since the condition $\,\nabla\hskip.7pt W\hskip-1pt=0$ implies
vanishing of the divergence of $\,W\hskip-.7pt$, which, in view of the second Bianchi
identity, is equivalent to the Co\-daz\-zi equation for $\,\sigma$, cf.\
\cite[formula (5.29) on p.\ 460]{dillen-verstraelen}.
\end{proof}
Assertion (i) in the next lemma is due to Ol\-szak \cite{olszak}.
\begin{lemma}\label{csnpd}Let\/ $\,\mathcal{D}\,$ be the Ol\-szak distribution\ of an essentially con\-for\-mal\-ly symmetric\ manifold\/
$\,(M,g)$, defined in the Introduction. Then
\begin{enumerate}
  \def{\rm\alph{enumi}}{{\rm\roman{enumi}}}
\item $\mathcal{D}\,$ is a null parallel distribution\ of dimension\/ $\,1\,$ or\/ $\,2$,
\item at every point\/ $\,x$, the space $\,\mathcal{D}_x$ contains the image of the
Ric\-ci tensor treated, with the aid of\/ $\,g_x$, as a linear operator
$\,{T\hskip-2pt_x\hskip-.9ptM}\to {T\hskip-2pt_x\hskip-.9ptM}\hskip-.7pt$,
\item $R(v,v\hskip.4pt'\hskip-1pt,\,\cdot\,,\,\cdot\,)
=\hskip.7pt W(v,v\hskip.4pt'\hskip-1pt,\,\cdot\,,\,\cdot\,)\hskip.7pt\,=\,\hs0\hskip.7pt\,$ whenever $\,v\,$ and
$\,v\hskip.4pt'$ are sections of $\,\dz^\perp\hskip-1pt$,
\item of the connections in the vector bundles $\,\mathcal{D}\hskip.7pt$ and\/
$\,\mathcal{E}=\dz^\perp\hskip-1pt/\hskip.4pt\mathcal{D}$, induced by the Le\-vi-Ci\-vi\-ta connection\ of\/ $\,g$, the latter is
always flat, and the former is flat if\/ $\,\mathcal{D}\hskip.7pt$ is one-di\-men\-sion\-al.
\end{enumerate}
\end{lemma}
\begin{proof}See \cite[Lemmas 2.1(ii) and 2.2]{derdzinski-roter-07}.
\end{proof}
For a $\,k\,$ times covariant tensor field $\,B\,$ on a pseu\-\hbox{do\hs-}Riem\-ann\-i\-an manifold\ $\,(M,g)$,
$\,k\ge1$, and a point $\,x\in M$, we denote by $\,\text{\rm Ker}\hskip1.5ptB_x$ the subspace of
$\,{T\hskip-2pt_x\hskip-.9ptM}\,$ formed by all vectors $\,v\,$ with
$\,B_x(v,\,\cdot\,,\ldots,\,\cdot\,)=0$. Its orthogonal complement
$\,(\text{\rm Ker}\hskip1.5ptB_x)^\perp$ is the {\it image} of $\,B_x$, that is, the subspace of
$\,{T\hskip-2pt_x\hskip-.9ptM}\,$ spanned by vectors $\,u\in {T\hskip-2pt_x\hskip-.9ptM}\,$ such that
$\,g(u,\,\cdot\,)=B(\,\cdot\,,u_2,\dots,u_k)\,$ for some
$\,u_2,\dots,u_k\in {T\hskip-2pt_x\hskip-.9ptM}$. If $\,B\,$ is parallel, the spaces $\,\text{\rm Ker}\hskip1.5ptB_x$ and
$\,(\text{\rm Ker}\hskip1.5ptB_x)^\perp$ form parallel distributions $\,\text{\rm Ker}\hskip1.5ptB\,$ and $\,(\text{\rm Ker}\hskip1.5ptB)^\perp$ on
$\,M$.

\begin{remark}\label{oldis}Given an essentially con\-for\-mal\-ly symmetric\ manifold\ $\,(M,g)\,$ such that
the Ol\-szak distribution\ $\,\mathcal{D}\,$ is $\,2$-di\-men\-sion\-al, we have
$\,W\hskip-.7pt=\hskip.7pt\varepsilon\hskip1pt\omega\hskip-.7pt\otimes\omega\,$ for some $\,\varepsilon=\pm\hs1\,$ and a
parallel differential $\,2$-form $\,\hskip.7pt\omega\hskip.7pt\,$ with
$\,\hskip.7pt\text{\rm rank}\,\hskip.7pt\omega\hskip.7pt=2$, defined, at each point of $\,M$, only up
to a sign. In addition,
\begin{equation}\label{owo}
\mathcal{D}\,=\,(\text{\rm Ker}\,\hskip.7pt\omega)^\perp\hskip-.7pt,
\end{equation}
In fact, if $\,x\in M\,$ and $\,u,v,v\hskip.4pt'$ are vector fields chosen so that
$\,u_x\in\mathcal{D}_x\smallsetminus\{0\}\,$ and $\,\hskip.7pt\varOmega_x\ne0$, for
$\,\hskip.7pt\varOmega\hskip.7pt=W(v,v\hskip.4pt'\hskip-1pt,\,\cdot\,,\,\cdot\,)$, then the $\,2$-form
$\,\varOmega_x$ is $\,\wedge$-di\-vis\-i\-ble by $\,\xi_{\hskip.7pt x}$, where
$\,\xi=g(u,\,\cdot\,)\,$ (cf.\ the definition of $\,\mathcal{D}$). Thus, if $\,\mathcal{D}\,$
is $\,2$-di\-men\-sion\-al, the image of the Weyl tensor $\,W_{\hskip-.7pt x}$ acting on exterior
$\,2$-forms is spanned by $\,\xi\wedge\xi\hskip.7pt'\hskip-1pt$, where
$\,\xi=g(u,\,\cdot\,)\,$ and $\,\xi\hskip.7pt'\hskip-.7pt=g(u\hskip.4pt'\hskip-1pt,\,\cdot\,)\,$ for
any basis $\,u,u\hskip.4pt'$ of $\,\dz\hskip-.3pt_x$. Since $\,W_{\hskip-.7pt x}$ acting on $\,2$-forms
is self-ad\-joint, our claim follows, (\ref{owo}) being immediate as
$\,(\text{\rm Ker}\,\hskip.7pt\omega)^\perp$ is the image of $\,\hskip.7pt\omega\hskip.4pt$.
\end{remark}
Next, at any point $\,x\,$ of any essentially con\-for\-mal\-ly symmetric\ manifold,
\begin{equation}\label{krp}
\mathrm{a)}\hskip12pt
(\text{\rm Ker}\hskip1pt{\rho}_x)^\perp\subset\hskip.7pt\mathcal{D}_x\subset\hskip.7pt\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp\subset\,\text{\rm Ker}\hskip1pt{\rho}_x\hskip.7pt,\hskip30pt
\mathrm{b)}\hskip12pt\mathcal{D}\,\subset\,\hskip.7pt\text{\rm Ker}\hskip2ptW\hskip.7pt,
\end{equation}
where $\,\mathcal{D}\,$ is the Ol\-szak distribution\ and $\,{\rho}\,$ denotes the Ric\-ci tensor. Namely,
the first inclusion in (a) follows from Lemma~\ref{csnpd}(ii) (as
$\,(\text{\rm Ker}\hskip1pt{\rho}_x)^\perp$ is the image of $\,{\rho}_x$), the second from
Lemma~\ref{csnpd}(i), and the third from the first. For (b), we consider the
two possible values $\,d\in\{1,2\}\,$ of the dimension of $\,\mathcal{D}\,$ (see
Lemma~\ref{csnpd}(i)). If $\,d=2$, we have $\,\mathcal{D}=(\text{\rm Ker}\hskip2ptW)^\perp\hskip-1pt$, that is,
$\,\mathcal{D}\,$ equals the image of $\,W\hskip.7pt$ (which coincides with the image
$\,(\text{\rm Ker}\,\hskip.7pt\omega)^\perp$ of $\,\hskip.7pt\omega\hskip.4pt$, cf. (\ref{owo})), while
$\,\mathcal{D}\subset\dz^\perp$ by Lemma~\ref{csnpd}(i). Now let $\,d=1$. As $\,W\hskip.7pt$ and
$\,\dz^\perp$ are both parallel, it suffices to establish (b) at any fixed
$\,x\in M\,$ with $\,{\rho}_x\ne0$. (Note that $\,g\,$ is not Ric\-ci-flat.) Now
the image $\,(\text{\rm Ker}\hskip1pt{\rho}_x)^\perp$ of $\,{\rho}_x$ is contained in $\,\text{\rm Ker}\hskip2ptW_{\hskip-.7pt x}$
according to \cite[Theorem 8(c) on p.\ 22]{derdzinski-roter-78}, while
$\,\mathcal{D}_x=(\text{\rm Ker}\hskip1pt{\rho}_x)^\perp$ by Lemma~\ref{csnpd}(ii), which yields (b).

\section{Proof of Theorem~\ref{pontr}, first part}\label{pfta}
The phrase `up to a factor' means, in this section, {\it up to a nonzero
constant factor, which may depend on the dimensions involved.}

Given a pseu\-\hbox{do\hskip.7pt-}Euclid\-e\-an inner product $\,\lg\,,\rg\,$ in
an oriented real vector space\ $\,V\hskip.7pt$ of even dimension $\,r\hskip.3pt=2m$, let $\,\varTheta\,$
be the volume form, with $\,\varTheta(e_1,\dots,e_r\hskip.3pt)=1\,$ for any
pos\-i\-tive-o\-ri\-ent\-ed orthonormal basis $\,e_1,\dots,e_r\hskip.3pt$ of
$\,V\hskip-1pt$. We denote by $\,\hskip.7pt\text{\rm Pf}\hskip.7pt\,$ the {\it Pfaffian}
function of $\,\lg\,,\rg$, assigning to an $\,m$-tuple of linear operators
$\,S_j:V\toV\hskip-1pt$, which are all skew-ad\-joint relative to $\,\lg\,,\rg$, the
value $\,s=\hskip.7pt\text{\rm Pf}\,(S_1,\dots,S_m)\in\text{\bf R}\,$ such that
$\,\zeta_1\hskip-1pt\wedge\ldots\wedge\zeta_m\hskip-.7pt=s\hskip.7pt\varTheta$, with the
$\,2$-forms $\,\zeta_j$ characterized by
$\,\zeta_j(u,v)=\langle S_ju,v\rangle\,$ for all $\,u,v\inV\hskip-1pt$.
\begin{lemma}\label{pfzer}For $\,S_j:V\toV\hskip.7pt$ as above,
$\,\hskip.7pt\text{\rm Pf}\,(S_1,\dots,S_m)=0\,$ if\/
$\,\bigcap_{\hskip.4pt j=1}^{\hskip.4pt m}\text{\rm Ker}\,S_j\ne\{0\}$.
\end{lemma}
\begin{proof}Our $\,\zeta_j$ are pull\-backs to $\,V\hskip.7pt$ of some $\,2$-forms
in the quotient space $\,V\hskip-1pt/V\hskip.4pt'\hskip-1pt$, where
$\,V\hskip.4pt'\hskip-1pt=\bigcap_{\hskip.4pt j=1}^{\hskip.4pt m}\text{\rm Ker}\,S_j$. Hence
$\,\zeta_1\hskip-1pt\wedge\ldots\wedge\zeta_m\hskip-.7pt=0\,$ if
$\,\dim\hskip.7pt(V\hskip-1pt/V\hskip.4pt'\hskip.4pt)<m=\dimV\hskip-.7pt$.
\end{proof}
Given an oriented real vector bundle $\,\mathcal{E}\hskip.7pt$ of fibre dimension
$\,r\hskip.3pt\ge1\,$ over a manifold $\,M$, let a pair $\,(\nabla,\gamma)\,$ consist of
a connection $\,\nabla\,$ and a $\,\nabla$-par\-al\-lel\
pseu\-\hbox{do\hskip.7pt-}Riem\-ann\-i\-an fibre metric $\,\gamma\,$ in $\,\mathcal{E}$.
The {\it Euler form} of $\,(\nabla,\gamma)\,$ then is the differential
$\,r\hskip.3pt$-form on $\,M\,$ equal to $\,0$, when $\,r\hskip.3pt\hskip.7pt$ is odd, and for even
$\,r\hskip.3pt\hskip.7pt$ obtained, up to a factor, by skew-sym\-met\-riza\-tion of the
$\,r\hskip.3pt\hskip.7pt$ times covariant tensor field that sends vector fields
$\,v_1,\dots,v_r\hskip.3pt$ to $\,\hskip.7pt\text{\rm Pf}\,
(R^\nabla\hskip-1pt(v_1,v_2),\dots,R^\nabla\hskip-1pt(v_{r\hskip.3pt-1},v_r\hskip.3pt))$, cf.\
(\ref{cvo}), with $\,\hskip.7pt\text{\rm Pf}\hskip.7pt\,$ as above for
$\,V\hskip-.7pt=\hskip.7pt\mathcal{E}_x$, $\,x\in M$.

The Euler form of $\,(\nabla,\gamma)\,$ is closed, and represents in cohomology
the real Euler class of the oriented bundle $\,\mathcal{E}$. See
\cite{avez-f}, \cite{chern}, \cite{law}, \cite{bell}.

Similarly, the real Pontryagin classes
$\,p_i(\mathcal{E})\in H^{4i}(M,\text{\bf R})\,$ of a real vector bundle
$\,\mathcal{E}\,$ over a manifold $\,M\,$ are the cohomology classes of the
Pontryagin forms of any connection $\,\nabla\,$ in $\,\mathcal{E}$, given by
explicit formulae involving the curvature tensor $\,R=R^\nabla\hskip-1pt$. To prove
vanishing of the Pontryagin forms (and classes) under some specific
assumptions, one may instead use what we call here the {\it generating forms},
the cohomology classes of which form another set of generators for the
Pontryagin algebra (the subalgebra of $\,H^*\hskip-1pt(M,\text{\bf R})\,$ generated by all
$\,p_i(\mathcal{E})$). The $\,i\hskip.7pt$th generating form of $\,\nabla$, for any
integer $\,i\ge1$, is the differential $\,4i$-form on $\,M\hskip-.7pt\,$ obtained, up
to a factor, by skew-sym\-met\-riza\-tion of the $\,4i\,$ times covariant
tensor field sending $\,v_1,\dots,v_{4i}$ to $\,\hskip.7pt\text{\rm tr}\,
[R^\nabla\hskip-1pt(v_1,v_2)\circ\ldots\circ R^\nabla\hskip-1pt(v_{4i-1},v_{4i})]$, with
$\,R^\nabla\hskip-1pt(v,v\hskip.4pt'\hskip.4pt)\,$ as in (\ref{cvo}). See \cite{chern-gcc}.

In the case where $\,\nabla\,$ is the Le\-vi-Ci\-vi\-ta connection\ of a pseu\-\hbox{do\hs-}Riem\-ann\-i\-an manifold\ $\,(M,g)\,$ and
$\,\mathcal{E}={T\hskip-.3ptM}$, we speak of the {\it Euler form\/} and {\it generating forms
of\/} $\,(M,g)$.

Theorem~\ref{pontr} (minus the claim about $\,\pi_1M$) is immediate from
the following lemma.
\begin{lemma}\label{eupoz}For an oriented essentially con\-for\-mal\-ly symmetric\ manifold\/ $\,(M,g)\,$ of
any dimension $\,n\ge4$, the Euler form and all Pontryagin forms are
identically zero.
\end{lemma}
\begin{proof}We fix a point $\,x\in M$, an integer $\,i\,$ with
$\,1\le i\le n/4$, where $\,n=\dim M$, and set
$\,b=\hskip.7pt\text{\rm tr}\,(B_1\circ\ldots\circ B_{2i})$, where
$\,B_j=W_{\hskip-.7pt x}(u_{2j-1},u_{2j}):{T\hskip-2pt_x\hskip-.9ptM}\to{T\hskip-2pt_x\hskip-.9ptM}\,$ for linearly independent
vectors $\,u_1,\dots,u_{4i}$ in $\,{T\hskip-2pt_x\hskip-.9ptM}$. (Notation as in (\ref{cvo}), with
the Weyl tensor $\,W$ instead of $\,R$.) For vanishing of the Pontryagin
forms, it suffices to prove that $\,b=0$, since, as shown by Avez
\cite{avez-c}, in the definition of generating forms of $\,(M,g)\,$ one
may replace the curvature tensor $\,R=R^\nabla$ by $\,W\hskip-.7pt$. We are thus
allowed to choose $\,x\,$ at which the Ric\-ci tensor $\,{\rho}_x$ is nonzero:
$\,W\hskip.7pt$ is parallel, and hence so is the $\,i\hskip.7pt$th generating form. For any
fixed $\,v\hskip.4pt'\hskip-.7pt\in{T\hskip-2pt_x\hskip-.9ptM}\,$ with $\,{\rho}_x(v\hskip.4pt'\hskip-1pt,\,\cdot\,)\ne0$,
Lemma~\ref{csnpd}(ii) implies that each $\,B_j$, when treated (with the aid of
$\,g_x$) as a $\,2$-form at $\,x$, is $\,\wedge$-di\-vis\-i\-ble by
$\,{\rho}_x(v\hskip.4pt'\hskip-1pt,\,\cdot\,)$. In other words,
$\,B_j:{T\hskip-2pt_x\hskip-.9ptM}\to{T\hskip-2pt_x\hskip-.9ptM}\,$ equals
$\,g_x(w_j,\,\cdot\,)\hskip.7pt v-{\rho}_x(v\hskip.4pt'\hskip-1pt,\,\cdot\,)\hskip.7pt w_j\,$ for some
$\,w_j\in{T\hskip-2pt_x\hskip-.9ptM}\,$ and the unique $\,v\in{T\hskip-2pt_x\hskip-.9ptM}\,$ with
$\,g_x(v,\,\cdot\,)={\rho}_x(v\hskip.4pt'\hskip-1pt,\,\cdot\,)$. Furthermore,
$\,\mathcal{D}_x\subset\text{\rm Ker}\hskip2ptW_{\hskip-.7pt x}\subset\hskip.7pt\text{\rm Ker}\,B_j$ and
$\,\mathcal{D}_x\subset\text{\rm Ker}\hskip1pt{\rho}_x$ (see (\ref{krp})), so that,
by Lemma~\ref{csnpd}(ii), $\,v\in\mathcal{D}_x$ and $\,w_j\in\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp$ (as $\,w_j$ lies
in the image $\,(\text{\rm Ker}\,B_j)^\perp$ of $\,B_j$). As $\,\mathcal{D}\,$ is null
(Lemma~\ref{csnpd}(i)),
$\,B_1\hskip-.7pt\circ B_2=-\hskip.7pt g_x(w_1,w_2){\rho}_x(v\hskip.4pt'\hskip-1pt,\,\cdot\,)\hskip.7pt v$,
$\,\hskip.7pt\text{\rm tr}\,(B_1\hskip-.7pt\circ B_2)=0\,$ and
$\,B_1\hskip-.7pt\circ B_2\hskip-.7pt\circ B_3=0\,$ if $\,i>1$, which implies that $\,b=0\,$
both for $\,i=1\,$ and $\,i>1$.

Now let $\,n\,$ be even. Given $\,x\in M$, we set
$\,s=\hskip.7pt\text{\rm Pf}\,(S_1,\dots,S_m)$, where $\,m=n/2\,$ and
$\,S_j=R_x(e_{2j-1},e_{2j}):{T\hskip-2pt_x\hskip-.9ptM}\to{T\hskip-2pt_x\hskip-.9ptM}\,$ for a basis $\,e_1,\dots,e_n$ of
$\,{T\hskip-2pt_x\hskip-.9ptM}\,$ containing a basis of $\,\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp$ (where $\,\mathcal{D}\,$ is the Ol\-szak distribution). To
obtain vanishing of the Euler form, we need to show that $\,s=0$. First,
$\,s=0\,$ if, for each $\,j\in\{1,\dots,m\}$, at least one of the vectors
$\,e_{2j-1},e_{2j}$ lies in $\,\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp\hskip-1pt$. In fact, Lemma~\ref{rwcod}(a) gives
$\,R_x(v,u\hskip.4pt'\hskip-1pt,u,\,\cdot\,)=0\,$ whenever $\,u\in\mathcal{D}_x$, $\,v\in\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp$ and
$\,u\hskip.4pt'\hskip-.7pt\in{T\hskip-2pt_x\hskip-.9ptM}$, as $\,\mathcal{D}_x\subset\text{\rm Ker}\hskip2ptW_{\hskip-.7pt x}$ and
$\,(g\wedge{\rho})_x(v,u\hskip.4pt'\hskip-1pt,u,\,\cdot\,)=0\,$ by (\ref{krp}); thus,
$\,\bigcap_{\hskip.4pt j=1}^{\hskip.4pt m}\text{\rm Ker}\,S_j$ contains the subspace
$\,\mathcal{D}_x\ne\{0\}\,$ (cf.\ Lemma~\ref{csnpd}(i)), and Lemma~\ref{pfzer} shows
that $\,s=0$.

In the remaining case, $\,e_{2j-1},e_{2j}\in\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp$ for some
$\,j\in\{1,\dots,m\}$. Namely, if $\,d\,$ denotes the dimension of $\,\mathcal{D}$,
then $\,\dz^\perp$ is $\,(2m-d)$-di\-men\-sion\-al, with $\,d\le2\le m\,$ by
Lemma~\ref{csnpd}(i). Among $\,e_1,\dots,e_{2m}$ there are
$\,2m-d\ge m\,$ elements of $\,\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp\hskip-1pt$, so that one of the
$\,m\,$ sets $\,\varSigma_j=\{e_{2j-1},e_{2j}\}\,$ must be contained in
$\,\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp$ (or else  $\,\varSigma_j\hskip-.7pt\cap\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp$ would, for each
$\,j=1,\dots,m$, have exactly one element, leading to a case which we already
excluded).

Now that $\,e_{2j-1},e_{2j}\in\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp$ for some $\,j$,
Lemma~\ref{csnpd}(iii) gives $\,S_j=0$, and hence $\,s=0$.
\end{proof}

\section{Proof of Theorem~\ref{parvf}}\label{pote}
In any essentially con\-for\-mal\-ly symmetric\ manifold\ $\,(M,g)\,$ such that the Ol\-szak distribution\ $\,\mathcal{D}\,$ is one-di\-men\-sion\-al,
setting $\,\mathcal{E}=\dz^\perp\hskip-1pt/\hskip.4pt\mathcal{D}$, one has a vec\-tor-bun\-dle morphism
\begin{equation}\label{phi}
\varPhi:(\mathcal{D}^*)^{\otimes2}\to\hskip.7pt(\mathcal{E}^*){}^{\otimes2}
\end{equation}
defined as follows. Given $\,x\in M\,$ and $\,\lambda,\lambda'\in\dz\hskip-.3pt_x{}\hskip-4.2pt^*$, we
declare
$\,\varPhi_x(\lambda\otimes\lambda'\hskip.4pt):\mathcal{E}_x\times\mathcal{E}_x\to\text{\bf R}\,$ to be the
symmetric bilinear form sending the cosets $\,v+\mathcal{D}_x$ and
$\,v\hskip.4pt'\hskip-.7pt+\mathcal{D}_x$ of vectors $\,v,v\hskip.4pt'\hskip-.7pt\in\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp$ to
$\,W_{\hskip-.7pt x}(v,u,u\hskip.4pt'\hskip-1pt,v\hskip.4pt'\hskip.7pt)$, where $\,u,u\hskip.4pt'\hskip-.7pt\in {T\hskip-2pt_x\hskip-.9ptM}\,$ are any
vectors with $\,\lambda=g_x(u,\,\cdot\,)\,$ and
$\,\lambda'\hskip-.7pt=g_x(u\hskip.4pt'\hskip-1pt,\,\cdot\,)\,$ on $\,\mathcal{D}_x$. Note that, as
$\,\mathcal{D}_x\subset\text{\rm Ker}\hskip2ptW_{\hskip-.7pt x}$ by (\ref{krp}.b), the value $\,W_{\hskip-.7pt x}(v,u,u\hskip.4pt'\hskip-1pt,v\hskip.4pt'\hskip.7pt)\,$
depends just on the $\,\mathcal{D}_x$-co\-sets, rather than the vectors $\,v,v\hskip.4pt'$
themselves, while, by Lemma~\ref{csnpd}(iii),
$\,W_{\hskip-.7pt x}(v,u,u\hskip.4pt'\hskip-1pt,v\hskip.4pt'\hskip.7pt)\,$ is not affected by how $\,u\,$ and
$\,u\hskip.4pt'$ were chosen: two such choices of either vector differ by an
element of $\,\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp\hskip-1pt$.
\begin{remark}\label{paral}If $\,(M,g)\,$ is essentially con\-for\-mal\-ly symmetric\ and $\,\mathcal{D}\,$ is one-di\-men\-sion\-al,
the Le\-vi-Ci\-vi\-ta connection\ of $\,g\,$ induces flat connections in both bundles
$\,(\mathcal{D}^*)^{\otimes2}$ and $\,(\mathcal{E}^*){}^{\otimes2}$ (by
Lemma~\ref{csnpd}(iv)), and the morphism $\,\varPhi\,$ is
\begin{enumerate}
  \def{\rm\alph{enumi}}{{\rm\alph{enumi}}}
\item parallel relative to those connections,
\item nonzero, and hence injective, at every point $\,x\in M$.
\end{enumerate}
Here (a) states that $\,\varPhi\,$ is parallel as a section of the bundle
$\,\hskip.7pt\text{\rm Hom}\hskip.7pt((\mathcal{D}^*)^{\otimes2}\hskip-1pt,\hskip.7pt(\mathcal{E}^*){}^{\otimes2}\hskip.4pt)\,$
with the induced flat connection, or, equivalently, that the
$\,\varPhi$-im\-age of any parallel local section of $\,(\mathcal{D}^*)^{\otimes2}$
is parallel in $\,(\mathcal{E}^*){}^{\otimes2}\hskip-1pt$.

Assertion (a) is obvious from naturality of $\,\varPhi$, since $\,W\hskip.7pt$ is
parallel. To verify (b), note that if we had $\,W_{\hskip-.7pt x}(v,u,u,v\hskip.4pt'\hskip.7pt)=0\,$
for a fixed $\,u\in{T\hskip-2pt_x\hskip-.9ptM}\smallsetminus\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp$ and all
$\,v,v\hskip.4pt'\hskip-.7pt\in\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp\hskip-1pt$, the components of $\,W_{\hskip-.7pt x}$ in a basis
consisting of $\,u\,$ and a basis of $\,\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp$ would all vanish (by
Lemma~\ref{csnpd}(iii)), even though $\,(M,g)\,$ is not conformally flat.
\end{remark}
\begin{remark}\label{fibme}Given an essentially con\-for\-mal\-ly symmetric\ manifold\ $\,(M,g)$, let $\,\mathcal{D}\,$ and
$\,\mathcal{E}\,$ be as in (\ref{phi}). Since $\,\mathcal{D}\,$ is the
$\,g$-null\-space subbundle of $\,\dz^\perp$ (cf.\ Lemma~\ref{csnpd}(i)), the
metric $\,g$, restricted to $\,\dz^\perp\hskip-1pt$, descends to a
pseu\-\hbox{do\hskip.7pt-}Riem\-ann\-i\-an fibre metric $\,\gamma\,$ on $\,\mathcal{E}$.
Clearly, $\,\gamma$ is parallel relative to the connection induced by the Le\-vi-Ci\-vi\-ta connection\
$\,\nabla\,$ of $\,g$. Being $\,\nabla$-par\-al\-lel, $\,\dz^\perp$ is integrable and has
totally geodesic leaves, and the Le\-vi-Ci\-vi\-ta connection\ of $\,g\,$ induces on each leaf a
tor\-sion\-free connection, which is flat in view of Lemma~\ref{csnpd}(iii).

If the sign pattern of $\,g\,$ is $\,(i_-,i_+)$, with $\,i_-$ minuses and
$\,i_+$ pluses, then $\,\gamma\,$ has the sign pattern $\,(i_--d,i_+-d)$,
where $\,d\,$ the dimension of the distribution $\,\mathcal{D}$. This is clear if
one chooses a subspace $\,V\hskip.7pt$ of $\,\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp$ with
$\,\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp\hskip-.7pt=\mathcal{D}_x\oplusV\hskip-1pt$, for any $\,x\in M$, and notes that
$\,{T\hskip-2pt_x\hskip-.9ptM}=V\hskip-1pt\oplusV^\perp\hskip-1pt$, while $\,V\hskip.7pt$ and $\,V^\perp$ have
the sign patterns equal to that of $\,\gamma\,$ and, respectively,
$\,(d,d)\,$ (as $\,\mathcal{D}_x\subsetV^\perp$).
\end{remark}
\begin{proof}[Proof of Theorem~{\rm\ref{parvf}}]Let $\,(M,g)\,$ be essentially con\-for\-mal\-ly symmetric\ and
Lo\-rentz\-i\-an, and let $\,d\,$ be the dimension of $\,\mathcal{D}$. Then
$\,d=1\,$ by Lemma~\ref{csnpd}(i), since $\,d\le1\,$ due to the
Lo\-rentz\-i\-an sign pattern \hbox{$-$\hskip1pt$+$\hskip1pt$\dots$\hskip1pt$+$}\hskip.7pt. (In fact, $\,{T\hskip-2pt_x\hskip-.9ptM}\,$ contains
$\,V\oplus\mathcal{D}_x$, for any $\,x\in M\,$ and any co\-di\-men\-sion-one
subspace $\,V\subset {T\hskip-2pt_x\hskip-.9ptM}\,$ on which $\,g_x$ is positive definite.)

The parallel injective morphism $\,\varPhi\,$ in (\ref{phi}) now gives rise to
a fibre norm $\,|\hskip2.3pt|\,$ in the line bundle $\,\mathcal{D}$, which is parallel
(invariant under parallel transports). Namely, for $\,\hskip.7pt x\in M$ and
$\,u\in\mathcal{D}_x\hskip-.7pt\smallsetminus\{0\}$, we set
$\,|u|=|\hskip.7pt\varPhi_x(\lambda\otimes\lambda)|^{-1/2}\hskip-1pt$, where
$\,\lambda\in\dz\hskip-.3pt_x{}\hskip-4.2pt^*$ is chosen so that $\,\lambda(u)=1$, and the latter
$\,|\hskip2.3pt|\,$ is the fibre norm in $\,(\mathcal{E}^*){}^{\otimes2}$
corresponding to the fibre metric $\,\gamma\,$ in $\,\mathcal{E}$. Note that $\,\gamma\,$ is
positive definite as $\,d=1\,$ (see Remark~\ref{fibme}). Since a
$\,|\hskip2.3pt|$-unit section of $\,\mathcal{D}$ is parallel, this proves
Theorem~\ref{parvf}.
\end{proof}

\section{Proof of Theorem~\ref{bdcir}}\label{potb}
Let an essentially con\-for\-mal\-ly symmetric\ manifold\ $\,(M,g)\,$ satisfy one of the conditions
\begin{enumerate}
  \def{\rm\alph{enumi}}{{\rm\roman{enumi}}}
\item $M\,$ is simply connected and the Ol\-szak distribution\ $\,\mathcal{D}\,$ is one-di\-men\-sion\-al, or
\item $g\,$ is Lo\-rentz\-i\-an and $\,\mathcal{D}\,$ is trivial as a real line
bundle, cf.\ Theorem~\ref{parvf}.
\end{enumerate}
Then there exist functions $\,\psi,\phi:M\to\text{\bf R}\,$ and a vector field
$\,u\,$ on $\,M\,$ such that
\begin{enumerate}
  \def{\rm\alph{enumi}}{{\rm\alph{enumi}}}
\item $u\,$ is parallel, nonzero, and spans $\,\mathcal{D}$,
\item the $\,1$-form $\,\xi=g(u,\,\cdot\,)\,$ is parallel, the Ric\-ci tensor
$\,{\rho}\,$ equals $\,\psi\,\xi\otimes\xi$, and $\,d\psi=\phi\,\xi$,
\item $\phi\,$ is nonconstant if $\,M\,$ is compact.
\end{enumerate}
Still assuming (i) or (ii), we define a vec\-tor-bun\-dle morphism
$\,A:\mathcal{E}\to\mathcal{E}\hskip.7pt$ by \hbox{requiring that} $\,\gamma_x(A_x\eta,\,\cdot\,)
=[\varPhi_x(\lambda\otimes\lambda)](\eta,\,\cdot\,)\,$ for $\,x\in M\,$ and
$\,\eta\in\mathcal{E}_x$ with $\,\lambda\in\mathcal{D}_x^*$ such that $\,\lambda(u_x)=1$.
(Notation of Lemma~\ref{csnpd}(iv), Remark~\ref{fibme} and (\ref{phi}).) Then
\begin{enumerate}
  \def{\rm\alph{enumi}}{{\rm\alph{enumi}}}
\item[(d)] $A\,$ is $\,\nabla$-par\-al\-lel\ as a section of
$\,\hskip.7pt\text{\rm Hom}\hskip.7pt(\mathcal{E},\mathcal{E})$, nonzero, self-ad\-joint relative to
$\,\gamma\,$ and traceless at every point,
\item[(e)] in the case where $\,A_x:\mathcal{E}_x\to\mathcal{E}_x$ has
$\,n-2=\dim\mathcal{E}_x$ distinct eigenvalues at some/every point
$\,x\in M$, for $\,n=\dim M\ge4$, the bundle $\,\mathcal{E}\hskip.7pt$ over $\,M\,$ is
an orthogonal direct sum of $\,\nabla$-par\-al\-lel\ real-line subbundles.
\end{enumerate}
Under the assumption (i), there exists a function $\,t:M\to\text{\bf R}\,$ such that,
for $\,\xi=g(u,\,\cdot\,)$,
\begin{enumerate}
  \def{\rm\alph{enumi}}{{\rm\alph{enumi}}}
\item[(f)] $\xi=\hskip.4pt dt\,$ (or, equivalently, $\,u=\nabla t$) and $\,\psi\,$ is,
locally, a function of $\,t$.
\end{enumerate}
In fact, $\,u\,$ in (a) exists in view of Lemma~\ref{csnpd}(iv) and
Theorem~\ref{parvf}, while (b), for some $\,\psi\,$ and $\,\phi$, follows from
Lemma~\ref{csnpd}(ii), since $\,\nabla\hskip-.7pt{\rho}=d\psi\otimes\xi\otimes\xi\,$ is
totally symmetric by Lemma~\ref{rwcod}(b). Now, if $\,\phi\,$ were constant
and $\,M\,$ compact,
$\,d\psi\,$ would be parallel (as $\,\xi\,$ is), and so, being zero
somewhere, $\,d\psi\,$ would vanish identically. However, $\,\psi\,$ is
nonconstant, since $\,{\rho}\,$ cannot be parallel: $\,g\,$ is conformally
symmetric but not locally symmetric. This contradiction proves (c). Next,
(d) holds in view of Remark~\ref{paral} and
Theorem~\ref{parvf}, with trace\-less\-ness of $\,A\,$ due to vanishing of
the contractions of $\,W\hskip-.7pt$. Assertion (e) is now immediate, the subbundles
in question being the eigenspace bundles of $\,A$. Finally, $\,\xi\,$ is
parallel, and hence closed, so that (b) implies (f).
\begin{proof}[Proof of Theorem~{\rm\ref{bdcir}}]Let $\hskip.7pt(M,g)\hskip.7pt$ be a compact
essentially con\-for\-mal\-ly symmetric\ Lo\-rentz\-i\-an manifold. Theorem~\ref{parvf} allows us to assume that
$\,(M,g)\,$ admits a global parallel vector field $\,u\,$ spanning the
one-di\-men\-sion\-al\ null parallel distribution\ $\,\mathcal{D}$. Condition (ii) above is therefore satisfied,
which implies (b), while the function $\,\phi\,$ in (b) is nonconstant
by (c). Our assertion is now immediate from Lemmas~\ref{clext}
and~\ref{csnpd}(iii).
\end{proof}
\begin{remark}\label{concl}Let $\,(M,g)\,$ be any compact essentially con\-for\-mal\-ly symmetric\
Lo\-rentz\-i\-an manifold\ such that the Ol\-szak distribution\ $\,\mathcal{D}\,$ is trivial as a real line
bundle. Choosing $\,\xi,\phi,\psi\,$ as in (a) -- (c), and $\,t\,$ as in (f)
(where, for $\,t\,$ to exist, we use instead of $\,(M,g)\,$ its universal
covering manifold\ $\,(\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\hskip.7pt)$), we see that $\,\xi,\phi,\psi\,$ and $\,t\,$
satisfy all the hypotheses of Lemma~\ref{clext}. Consequently, {\it they
satisfy the conclusions of Lemma~{\rm\ref{clext}} as well}. This proves
the claims immediately following Theorem~\ref{nolor} in the Introduction.
\end{remark}

\section{Examples}\label{xpls}
Suppose that we are given a nonconstant $\,C^\infty$ function
$\,f:\text{\bf R}\to\text{\bf R}\hskip.7pt$, a real vector space\ $\,V\hskip.7pt$ of dimension $\,n-2\ge2\,$ with a
pseu\-\hbox{do\hskip.7pt-}Euclid\-e\-an inner product $\,\lg\,,\rg$, and a nonzero
traceless linear operator $\,A:V\toV\hskip-.7pt$, self-ad\-joint relative to
$\,\lg\,,\rg$. Following \cite{roter}, we use such data to define a pseu\-\hbox{do\hs-}Riem\-ann\-i\-an metric\
$\,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt=\kappa\,dt^2\hskip.7pt+\,dt\,ds\,+\,h\,$ on the manifold
$\,\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt=\mathbf{R}\hskip-.5pt^2\hskip-.7pt\timesV\hskip-.7pt$, dif\-feo\-mor\-phic\ to $\,{\mathbf{R}}^{\hskip-.6ptn}\hskip-1pt$, where products of
differentials stand for symmetric products, $\,t,s\,$ are the Cartesian
coordinates on the $\,\mathbf{R}\hskip-.5pt^2$ factor, $\,h\,$ denotes the pull\-back to
$\,\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\,$ of the flat pseu\-\hbox{do\hs-}Riem\-ann\-i\-an metric\ on $\,V$ corresponding to the inner product
$\,\lg\,,\rg$, and $\,\kappa:\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\to\text{\bf R}\,$ is given by
$\,\kappa(t,s,v)=f(t)\hskip.4pt\langle v,v\rangle+\langle Av,v\rangle$.

Let $\,\mathcal{E}$ be the vector space of all $\,C^\infty$ solutions
$\,u:\text{\bf R}\toV\,$ to the differential equation
$\,\ddot u(t)=f(t)\hskip.4pt u(t)+Au(t)$, and let $\,\text{\bf P}\,$ be the additive group
of all $\,p\in\text{\bf R}\,$ with $\,f(t+p)=f(t)$ for every real $\,t$. The set
$\,\mathrm{G}=\text{\bf P}\times\text{\bf R}\times\mathcal{E}\,$
has a unique group structure such that the formula
$\,(p,q,u)\cdot(t,s,v)=(t+p,\hskip.7pt s+q-\langle\dot u(t),2v+u(t)\rangle,\hskip.7pt v+u(t))$, for
$\,(p,q,u)\in\mathrm{G}\,$ and $\,(t,s,v)\in\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt=\mathbf{R}\hskip-.5pt^2\hskip-.7pt\timesV\hskip-.7pt$, describes a
group action of $\,\mathrm{G}\hskip.7pt$ on $\,\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt$. (See
\cite[Section 2]{derdzinski-roter}.)
\begin{lemma}\label{ecsrr}For any choice of the above data
$\,f,n,V\hskip-1pt,\lg\,,\rg\,$ and\/ $\,A$,
\begin{enumerate}
  \def{\rm\alph{enumi}}{{\rm\roman{enumi}}}
\item the metric\/ $\,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\,$ is essentially con\-for\-mal\-ly symmetric,
\item the sign pattern of\/ $\,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\,$ arises from that of\/ $\,\lg\,,\rg\,$ by
adding one plus and one minus,
\item the group\/ $\,\mathrm{G}\hskip.7pt$ acts on\/ $\,(\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\hskip.7pt)\,$ by isometries,
\item if\/ $\,n=4\,$ and the metric $\,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\,$ is Lo\-rentz\-i\-an,
\begin{enumerate}
  \def{\rm\alph{enumi}}{{\rm\alph{enumi}}}
\item[a)] $\mathrm{G}\hskip.7pt$ is a subgroup of finite index in the full isometry group
of\/ $\,(\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\hskip.7pt)$,
\item[b)] $(\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\hskip.7pt)\hskip.7pt$ is not the universal covering space of any
compact pseu\-\hbox{do\hs-}Riem\-ann\-i\-an manifold.
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{proof}For (i), see \cite[Theorem 3]{roter} or
\cite[Lemma 2.1]{derdzinski-roter}, while (ii) is obvious, and (iii) is
immediate from \cite[Lemma 2.2]{derdzinski-roter}.

Generally, the index $\,\hskip.7pt{\rm ind}\hskip.7pt(\mathrm{G}\hskip.7pt'\hskip-1pt,\mathrm{G})\,$ of a subgroup
$\,\mathrm{G}\,$ in a group $\,\mathrm{G}\hskip.7pt'$ is the cardinality of the quotient set
$\,\mathrm{G}\hskip.7pt'\hskip-1pt/\hskip.7pt\mathrm{G}\,$ consisting of all left co\-sets of $\,\mathrm{G}\,$ in
$\,\mathrm{G}\hskip.7pt'\hskip-1pt$. If $\,\mathrm{H}\hskip.7pt'\hskip-.7pt\subset\mathrm{G}\hskip.7pt'$ is a subgroup such that
$\,\mathrm{G}\hskip.7pt'\hskip-.7pt=\mathrm{H}\hskip.7pt'\mathrm{G}$, then, for $\,\mathrm{H}=\mathrm{G}\cap\mathrm{H}\hskip.7pt'\hskip-1pt$, the
inclusion mapping $\,\mathrm{H}\hskip.7pt'\hskip-.7pt\to\mathrm{G}\hskip.7pt'$ clearly induces a bijection
$\,\mathrm{H}\hskip.7pt'\hskip-1pt/\hskip.7pt\mathrm{H}\,\to\,\mathrm{G}\hskip.7pt'\hskip-1pt/\hskip.7pt\mathrm{G}$, and so
$\,\hskip.7pt{\rm ind}\hskip.7pt(\mathrm{G}\hskip.7pt'\hskip-1pt,\mathrm{G})=\hskip.7pt{\rm ind}\hskip.7pt(\mathrm{H}\hskip.7pt'\hskip-1pt,\mathrm{H})$.
Here are two special cases in which groups $\,\mathrm{G}\hskip.7pt'\hskip-1pt,\mathrm{G}\,$ and
$\,\mathrm{H}\,$ satisfy the above assumption, and hence the conclusion:
\begin{enumerate}
  \def{\rm\alph{enumi}}{{\rm\roman{enumi}}}
\item[(I)] $\mathrm{H}\hskip.7pt'\hskip-.7pt=\hskip.7pt\text{\rm Ker}\,\varPi\,$ for a group homomorphism
$\,\varPi:\mathrm{G}\hskip.7pt'\hskip-.7pt\to\mathrm{K}\,$ with $\,\varPi(\mathrm{G})=\mathrm{K}\hskip.7pt$,
\item[(II)] $\mathrm{G}\hskip.7pt'$ acts from the left on a set $\,Y\hskip-.7pt$, the action
restricted to $\,\mathrm{G}\hskip.7pt$ is transitive, and $\,\mathrm{H}\hskip.7pt'$ is the isotropy
subgroup of $\,\mathrm{G}\hskip.7pt'$ at some $\,y\in Y\hskip-.7pt$.
\end{enumerate}
Let $\,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\,$ now be Lo\-rentz\-i\-an, and let $\,\mathrm{G}\hskip.7pt'$ denote the group
of those isometries of $\,\hskip-.7pt(\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\hskip.7pt)\hskip.7pt$
which preserve the $\,1$-form $\,dt$. The Ric\-ci tensor of $\,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\,$ is given
by $\,{\rho}=(2-n)\hsf(t)\,dt\otimes dt$, and $\,dt\,$ is parallel. (See
\cite[p.\ 93]{roter}, where the sign convention for $\,{\rho}\,$ is the opposite
of ours.) Thus, by Lemma~\ref{csnpd}(ii), the Ol\-szak distribution\ $\,\mathcal{D}\,$ is spanned by the
null parallel vector field $\,u=\nabla t$. Since $\,u\,$ can be naturally
normalized with the aid of a parallel fibre norm in $\,\mathcal{D}\,$ (see the end of
Section~\ref{pote}), isometries of $\,(\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\hskip.7pt)\,$ leave $\,dt\,$ invariant
up to a sign, so that the full isometry group of $\,(\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\hskip.7pt)\,$ contains
$\,\mathrm{G}\hskip.7pt'$ as a subgroup of index at most $\,2$, and (iv-a) will follow if we
show that $\,\hskip.7pt{\rm ind}\hskip.7pt(\mathrm{G}\hskip.7pt'\hskip-1pt,\mathrm{G})\,$ is finite.

As $\,\mathrm{G}\hskip.7pt'$ preserves $\,dt\,$ and $\,{\rho}=(2-n)\hsf(t)\,dt\otimes dt$, we
have $\,t\circ\alpha=t+\varPi(\alpha)\,$ for some homomorphism
$\,\varPi:\mathrm{G}\hskip.7pt'\hskip-.7pt\to\text{\bf R}\,$ and all $\,\alpha\in\mathrm{G}\hskip.7pt'\hskip-1pt$, while the
function $\,f(t):\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\to\text{\bf R}\,$ is $\,\mathrm{G}\hskip.7pt'\hskip-.7pt$-in\-var\-i\-ant. Thus,
$\,\varPi(\mathrm{G}\hskip.7pt'\hskip.7pt)\,$ coincides with the additive group $\,\text{\bf P}$ defined
earlier in this section, and so
$\,\hskip.7pt{\rm ind}\hskip.7pt(\mathrm{G}\hskip.7pt'\hskip-1pt,\mathrm{G})
=\hskip.7pt{\rm ind}\hskip.7pt(\text{\rm Ker}\,\varPi,\mathrm{G}\cap\hskip.7pt\text{\rm Ker}\,\varPi)$,
in view of (I) above for $\,\mathrm{K}=\text{\bf P}\,$ and $\,\varPi:\mathrm{G}\hskip.7pt'\hskip-.7pt\to\text{\bf P}$.
Next, for any fixed $\,t\in\text{\bf R}\hskip.7pt$, the action of
$\,\hskip.7pt\text{\rm Ker}\,\varPi\,$ leaves the affine subspace
$\,\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.5pt}_t=\{t\}\times\text{\bf R}\timesV\hskip.7pt$ of $\,\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\,$ invariant, and the
restriction of this action to
$\,\mathrm{G}\cap\hskip.7pt\text{\rm Ker}\,\varPi$ is transitive on $\,\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.5pt}_t$, since it
consists of affine transformations realizing all translational parts in
$\,\{0\}\times\text{\bf R}\timesV\hskip-.7pt$. Consequently, (II) gives
$\,\hskip.7pt{\rm ind}\hskip.7pt(\mathrm{G}\hskip.7pt'\hskip-1pt,\mathrm{G})=\hskip.7pt{\rm ind}\hskip.7pt(\mathrm{H}\hskip.7pt'\hskip-1pt,\mathrm{H})$,
where $\,\mathrm{H}\hskip.7pt'$ is the isotropy subgroup of $\,\mathrm{G}\hskip.7pt'$ at any fixed
$\,x\in\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt$, and $\,\mathrm{H}=\mathrm{G}\cap\mathrm{H}\hskip.7pt'\hskip-1pt$.

On the other hand, $\,\hskip.7pt{\rm ind}\hskip.7pt(\mathrm{H}\hskip.7pt'\hskip-1pt,\mathrm{H})\le4\,$ if $\,n=4$.
Namely, the infinitesimal action of $\,\mathrm{H}\hskip.7pt'$ on
$\,{T\hskip-2pt_x\hskip-.9pt\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt}=\mathbf{R}\hskip-.5pt^2\hskip-.7pt\timesV\hskip.7pt$ preserves $\,dt_x$ and the vector
$\,u_x=(0,1/2,0)$, along with the subspace $\,u_x^\perp\hskip-.7pt=\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp\hskip-1pt$. Hence
the action descends to $\,\mathcal{E}_x=\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp\hskip-1pt/\mathcal{D}_x$, where it preserves the inner
product $\,\gamma_x$ (see Remark~\ref{fibme}) and commutes with the operator
$\,A_x:\mathcal{E}_x\to\mathcal{E}_x$ which, under the obvious identification
$\,\mathcal{E}_x\approx\hsV\hskip-1pt$, coincides with our $\,A:V\toV\hskip-.7pt$. (See
\cite[the description of $\,W\hskip.7pt$ on p.\ 93]{roter}.) The differentials at
$\,x\,$ of elements of $\,\mathrm{H}\hskip.7pt'\hskip-1pt$, acting on
$\,{T\hskip-2pt_x\hskip-.9pt\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt}=\mathbf{R}\hskip-.5pt^2\hskip-.7pt\timesV\hskip-1pt$, thus have the form
$\,(t,s,v)\mapsto(t,s+\varphi(v),Lv)$, where $\,L:V\toV\hskip.7pt$ is a linear
isometry commuting with $\,A$, and $\,\varphi\inV\hskip.4pt^*\hskip-1pt$. As elements of
the subgroup $\,\mathrm{H}\,$ realize all $\,\varphi\inV\hskip.4pt^*\hskip-1pt$, and have
$\,L=\hskip.7pt\text{\rm Id}\hskip.4pt$, it follows that
$\,\hskip.7pt{\rm ind}\hskip.7pt(\mathrm{H}\hskip.7pt'\hskip-1pt,\mathrm{H})\le4\,$ (see Remark~\ref{cmiso} below),
which yields (iv-a).

Finally, to prove (iv-b), we may suppose that, on the contrary, some group
$\,\Gamma\,$ of isometries of $\,(\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\hskip.7pt)\,$ acts on $\,\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\,$ properly
dis\-con\-tin\-u\-ous\-ly, producing a compact quotient manifold. The same
is then true for the subgroup $\,\Gamma\cap\mathrm{G}\,$ of $\,\Gamma\,$ (as
$\,\Gamma\cap\mathrm{G}\,$ is of finite index in $\,\Gamma$, by (iv-a)), which in turn
contradicts \cite[Theorem~7.3]{derdzinski-roter}. Note that periodicity
of $\,f\,$ as a function of $\,t$, required in \cite{derdzinski-roter},
follows from Lemma~\ref{clext}(b), cf.\ Remark~\ref{concl}.
\end{proof}
\begin{remark}\label{cmiso}For a nonzero traceless self-ad\-joint linear
endomorphism $\,A\,$ of a pseu\-\hbox{do\hskip.7pt-}Euclid\-e\-an plane $\,V\hskip-.7pt$,
there exist at most four linear isometries $\,L:V\toV\hskip.7pt$ commuting with
$\,A$. This is clear when $\,A\,$ is di\-ag\-o\-nal\-izable, since $\,L\,$
must then send an orthonormal basis $\,(v,w)\,$ diagonalizing $\,A\,$ to
$\,(\pm\hskip.7pt v,\pm\hskip.7pt w)\,$ or $\,(\pm\hskip.7pt v,\mp\hskip.7pt w)$. On the other hand, if a
linear isometry $\,L\,$ commutes with $\,A\,$ and $\,A\,$ is
non-di\-ag\-o\-nal\-izable (so that $\,V\hskip.7pt$ is Lo\-rentz\-i\-an), we have
$\,L=\pm\hskip.7pt\text{\rm Id}\hskip.4pt$. In fact, let $\,L\ne\pm\hskip.7pt\text{\rm Id}\hskip.4pt$.
The two null lines in $\,V\hskip.7pt$ are interchanged by $\,L$ (if they were
preserved, $\,L\,$ would be di\-ag\-o\-nal\-izable, implying the same for
$\,A$). However, choosing a basis $\,(v,w)\,$ of null vectors with $\,Lv=w\,$
and $\,Lw=v$, we would then again diagonalize $\,L\,$ (and hence $\,A$),
this time with the eigenvectors $\,v\pm w$.
\end{remark}

\section{A classification theorem}\label{clth}
In the following theorem, $\,t\,$ denotes any fixed function $\,\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\to\text{\bf R}\,$
such that $\,u=\nabla t\,$ is a global parallel vector field spanning the
Ol\-szak distribution\ $\,\mathcal{D}$. Such $\,t\,$ exists according to (f) in Section~\ref{potb},
cf.\ Theorem~\ref{parvf}.
\begin{theorem}\label{class}Let\/ $\,(\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\hskip.7pt)\,$ be a simply connected
essentially con\-for\-mal\-ly symmetric\ Lo\-rentz\-i\-an manifold\ of dimension $\,n\ge4\,$ such that the leaves of
the parallel distribution $\,\dz^\perp$ are all complete and the function
$\,t:\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\to\text{\bf R}\,$ satisfies condition\/ {\rm(c)} in Lemma~{\rm\ref{clext}}.
Then, up to an isometry, $\,(\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\hskip.7pt)\,$ is one of the manifolds
constructed in Section\/~{\rm\ref{xpls}}.
\end{theorem}
Our proof of Theorem~\ref{class}, given in Section~\ref{potc}, uses the facts
presented below.

Let $\,(\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\hskip.7pt)\,$ be a simply connected essentially con\-for\-mal\-ly symmetric\ manifold\ such that the Ol\-szak distribution\
$\,\mathcal{D}\,$ is one-di\-men\-sion\-al. If $\,v,v\hskip.4pt'$ are sections of $\,\dz^\perp$ and
$\,u\,$ is a fixed nonzero parallel section of $\,\mathcal{D}$, while $\,u\hskip.4pt'$ is
any vector field, then
\begin{equation}\label{ruv}
R(u\hskip.4pt'\hskip-1pt,v)\hskip.4pt v\hskip.4pt'\,
=\,g(u\hskip.4pt'\hskip-1pt,u)\hskip.7pt[\hskip.7pt\gamma(A\underline{v\hskip-.8pt}\hskip.8pt,\underline{v\hh'\hskip-.8pt}\hskip.8pt\hskip.7pt)+f g(v,v\hskip.4pt'\hskip.4pt)\hskip.4pt]\hskip.7pt u\hskip.7pt,
\end{equation}
where $\,f:\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\to\text{\bf R}\,$ is given by $\,f=(2-n)^{-1}\psi$, for
$\,n=\dim\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\ge4$, with $\,\psi\,$ and $\,A\,$ defined as in
Section~\ref{potb}, and $\,\underline{v\hskip-.8pt}\hskip.8pt\,$ denotes the image of $\,v\,$ under the
quo\-tient-pro\-jec\-tion morphism $\,\dz^\perp\hskip-1pt\to\mathcal{E}=\dz^\perp\hskip-1pt/\hskip.4pt\mathcal{D}$. (Thus,
$\,f\,$ is characterized by $\,{\rho}=(2-n)\hsf\,\xi\otimes\xi$, for the
parallel $\,1$-form $\,\xi=g(u,\,\cdot\,)=\hskip.4pt dt$.)

In fact, by Lemma~\ref{csnpd}(iii), $\,R(u\hskip.4pt'\hskip-1pt,v)\hskip.4pt v\hskip.4pt'$ is orthogonal
to $\,\dz^\perp\hskip-1pt$, and hence equals a function times $\,u$. Since both
sides of (\ref{ruv}) are linear in $\,u\hskip.4pt'\hskip-1pt$, (\ref{ruv}) will follow if,
under the assumption $\,g(u\hskip.4pt'\hskip-1pt,u)=1$, applying
$\,g(u\hskip.4pt'\hskip-1pt,\,\cdot\,)\,$ to both sides we obtain the same value. This
last conclusion is in turn immediate from Lemma~\ref{rwcod}(a) combined with
the definition of $\,A\,$ in Section~\ref{potb} and the relation
$\,{\rho}=(2-n)\hsf\,\xi\otimes\xi$. (By (\ref{krp}.a),
$\,{\rho}\hskip.4pt(v,\,\cdot\,)={\rho}\hskip.4pt(v\hskip.4pt'\hskip-1pt,\,\cdot\,)=0$.)
\begin{remark}\label{param}Let $\,(\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\hskip.7pt)\,$ satisfy the assumptions of
Theorem~\ref{class}. We say that a curve $\,I\to\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt$, defined on an interval
$\,I\subset\text{\bf R}\hskip.7pt$, is {\it parametrized by the function\/}
$\,t:\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\to\text{\bf R}\,$ (chosen at the beginning of this section) if $\,t\,$ sends
the image of the curve dif\-feo\-mor\-phi\-cal\-ly\ onto $\,I$. Such a curve may be written as
$\,I\ni t\mapsto y(t)\in\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt$, with $\,t\,$ serving as the parameter.
\begin{enumerate}
  \def{\rm\alph{enumi}}{{\rm\alph{enumi}}}
\item Up to an affine re-pa\-ram\-e\-tri\-za\-tion, every geodesic in
$\,(\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\hskip.7pt)$, not tangent to the distribution $\,\dz^\perp\hskip-1pt$, is
parametrized by the function $\,t$.
\item If a curve $\,\text{\bf R}\ni t\mapsto y(t)\in\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\,$ is parametrized by the
function $\,t$, then so is every curve $\,t\mapsto x(t,s)\,$ in the variation
given by $\,x(t,s)=\hskip.7pt\exp_{\hskip.7pt y(t)}sw(t)$, where
$\,\text{\bf R}\ni t\mapsto w(t)\in\mathcal{D}_{y(t)}^\perp$ is any vector field along the
original curve, tangent to $\,\dz^\perp\hskip-1pt$.
\item For any curve $\,\text{\bf R}\ni t\mapsto y(t)\in\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\,$ is parametrized by the
function $\,t$, we have $\,g(\dot y,u)=1$, for $\,u=\nabla t$, and
$\,\nabla_{\!\dot y}\dot y\,$ is tangent to $\,\dz^\perp\hskip-1pt$.
\end{enumerate}
In fact, (a) and (b) follow since $\,\nabla dt=0$, so that $\,t\,$ restricted
to any geodesic is an affine function of the geodesic parameter, while the
leaves of $\,\dz^\perp$ are totally geodesic (Remark~\ref{fibme}), and $\,t\,$
is constant on each leaf as $\,\dz^\perp\hskip-.7pt=\hskip.7pt\text{\rm Ker}\,dt$. Now (c)
is immediate as $\,\dz^\perp\hskip-.7pt=u^\perp$ and $\,u=\nabla t\,$ is parallel:
differentiating $\,g(\dot y,u)=1$, we get $\,g(\nabla_{\!\dot y}\dot y,u)=0$.
\end{remark}
The following lemma is a crucial step in proving Theorem~\ref{class}.
\begin{lemma}\label{compl}Under the assumptions of Theorem~{\rm\ref{class}},
$\,(\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\hskip.7pt)\,$ is complete.
\end{lemma}
\begin{proof}Using (c) in Lemma~\ref{clext}, we may fix a curve
$\,\text{\bf R}\ni t\mapsto y(t)\in\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\,$ parametrized by the function $\,t\,$ (cf.\
Remark~\ref{param}), and consider the differential equation
\begin{equation}\label{ode}
\nabla_{\!\dot y}\nabla_{\!\dot y}w+R(\dot y,w)\hskip.4pt\dot y
+\nabla_{\!\dot y}\dot y\,=\,-\hskip.7pt Q(w)\hskip.4pt u/4
\end{equation}
imposed on vector fields $\,w\,$ along the curve which are tangent to
$\,\dz^\perp\hskip-1pt$. Here $\,u=\nabla t\,$ and
$\,Q(w)=3\hskip.7pt[\gamma(A\underline{w\hskip-.8pt}\hskip.8pt,\hskip.7pt\underline{w\hskip-.8pt}\hskip.8pt)]\dot{\,}\hskip-.7pt
+3f\hskip.7pt[g(w,w)]\dot{\,}\hskip-.7pt+2\dotf g(w,w)$, with $\,f,A,\underline{w\hskip-.8pt}\hskip.8pt\,$ as in
(\ref{ruv}), and $\,[\hskip4pt]\dot{\,}\hskip-.7pt=\hskip.4pt d/dt$. (Thus,
$\,\dotf=\hhd\hskip-.8ptf(y(t))/dt$.) By Remark~\ref{param}(c), both sides of
(\ref{ode}) are tangent to $\,\dz^\perp\hskip-1pt$, that is, orthogonal to $\,u$, as
$\,\nabla u=0\,$ and so $\,R(\,\cdot\,,\,\cdot\,,\,\cdot\,,u)=0$.

Every solution $\,w\,$ to (\ref{ode}) can be defined on the whole real line.
Namely, this is true, due to linearity, for solutions $\,\widetilde w\,$
(tangent to $\,\dz^\perp$) of the equation
$\,\nabla_{\!\dot y}\nabla_{\!\dot y}\widetilde w
+R(\dot y,\widetilde w\hskip.7pt)\hskip.4pt\dot y+\nabla_{\!\dot y}\dot y=0$. Using any such
$\,\widetilde w\,$ and any function $\,\mu:\text{\bf R}\to\text{\bf R}\,$ with the second
derivative $\,\ddot\mu=Q(\widetilde w\hskip.7pt)/4$, we now get a solution
$\,w=\widetilde w-\mu u\,$ to (\ref{ode}), defined on $\,\text{\bf R}\hskip.7pt$. (Note
that $\,Q(w)=Q(\widetilde w\hskip.7pt)$, and $\,R(\dot y,u)\hskip.4pt\dot y=0\,$ since
$\,\nabla u=0$.)

Any solution $\,w\,$ to (\ref{ode}), defined on $\,\text{\bf R}\hskip.7pt$, leads to the
variation of curves in $\,\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\,$ given by $\,x(t,s)=\hskip.7pt\exp_{\hskip.7pt y(t)}sw(t)$.
Let $\,v\,$ be the vector field along the variation such that $\,v_s=0\,$ for
all $\,(t,s)$, and $\,v=\nabla_{\!\dot y}\dot y\,$ at $\,s=0\,$ (notation as
in (\ref{xts})). We have
\begin{equation}\label{xtt}
\mathrm{i)}\hskip9ptx_{tt}\,+\,(s-1)[v-s\hskip.7pt Q(x_s)\hskip.4pt u/4\hskip.4pt]\,=\,0\hskip.7pt,
\hskip29pt\mathrm{ii)}\hskip9pt[Q(x_s)]_s\,=\,\hs0\hskip.7pt.
\end{equation}
where the subscripts now also stand for partial derivatives of functions of
$\,(t,s)$, and $\,Q(x_s)=3\hskip.7pt[\gamma(A\underline{x_s\hskip-.8pt}\hskip.8pt,\hskip.7pt\underline{x_s\hskip-.8pt}\hskip.8pt)]_t
+3f\hskip.7pt[g(x_s,x_s)]_t+2f_t\hskip.7pt g(x_s,x_s)$. Before proving (\ref{xtt}), note
that
\begin{equation}\label{uxo}
\mathrm{a)}\hskip7ptg(x_t,u)=1\hskip.7pt,
\hskip9pt\mathrm{b)}\hskip7ptx_{ss}=0\hskip.7pt,
\hskip9pt\mathrm{c)}\hskip7ptx_s,x_{tt}\text{\rm\ and\ }\,x_{st}=x_{ts}
\text{\rm\ are\ all\ tangent\ to\ }\,\dz^\perp.
\end{equation}
In fact, (\ref{uxo}.a) and the claim about $\,x_{tt}$ in (\ref{uxo}.c) are
immediate from clauses (b), (c) in Remark~\ref{param}. Next, $\,x_{ss}=0\,$
and $\,x_s$ is tangent to $\,\dz^\perp\hskip-1pt$, as $\,\dz^\perp$ has totally geodesic
leaves (Remark~\ref{fibme}), while the curves $\,s\mapsto x(t,s)\,$ are
geodesics, tangent to $\,\dz^\perp$ at $\,s=0$. Finally, $\,x_{st}$ is tangent to
$\,\dz^\perp\hskip-1pt$, since so is $\,x_s$ and $\,\dz^\perp$ is parallel.

Furthermore, for the covariant derivatives $\,R_t,R_s$ of the curvature tensor,
\begin{equation}\label{fso}
\mathrm{a)}\hskip14ptR_t(x_t,x_s)x_s\,=\,f_t\hskip.7pt g(x_s,x_s)\hskip.7pt u\hskip.7pt,
\hskip22pt\mathrm{b)}\hskip14ptR_s\,=\,\hs0\hskip.7pt,
\hskip22pt\mathrm{c)}\hskip14ptf_s\,=\,\hs0\hskip.7pt.
\end{equation}
Namely, by (f) in Section~\ref{potb}, $\,f\,$ is a function of $\,t$, while
$\,t_s=0\,$ as the curves $\,s\mapsto x(t,s)$ are tangent to
$\,\dz^\perp\hskip-.7pt=\hskip.7pt\text{\rm Ker}\,dt$, and (\ref{fso}.c) follows. Since
$\,{\rho}=(2-n)\hsf(t)\,dt\otimes dt$, $\,\nabla W=0$ and $\,\nabla dt=0$,
(\ref{fso}.a) and (\ref{fso}.b) are immediate from (\ref{fso}.c) and
Lemma~\ref{rwcod}(a).

We can now prove (\ref{xtt}). Relation (\ref{xtt}.ii) is obvious from
(\ref{fso}.c), as $\,g,\gamma\,$ and $\,A\,$ are parallel, so that (\ref{uxo}.b)
yields $\,[\gamma(A\underline{x_s\hskip-.8pt}\hskip.8pt,\hskip.7pt\underline{x_s\hskip-.8pt}\hskip.8pt)]_s=[g(x_s,x_s)]_s=0$. Denoting by $\,\widetilde v\,$ the left-hand side of
(\ref{xtt}.i), we get $\,\widetilde v=\widetilde v_s=0\,$ at $\,s=0\,$ (from
(\ref{xts}.a), (\ref{ode}) and (\ref{xtt}.ii) with $\,u_s=v_s=0$). Finally,
$\,\widetilde v_{ss}=0\,$ for all $\,(t,s)$, since (\ref{xts}.b) and
(\ref{ruv}) give $\,x_{ttss}=Q(x_s)\hskip.4pt u/2$. More precisely, according to
(\ref{xts}.b) with $\,x_{ss}=0\,$ (see (\ref{uxo}.b)), $\,x_{ttss}$ equals the
sum of three curvature terms, so that, using the Leibniz rule with
$\,x_{ts}=x_{st}$, we obtain
$\,x_{ttss}=3\hskip.4pt R(x_t,x_s)x_{st}+R_t(x_t,x_s)x_s$, all the other terms being
zero as a consequence of Lemma~\ref{csnpd}(iii) combined with
(\ref{uxo}.b,\hskip.7pt c), and (\ref{fso}.b). Using (\ref{ruv}) with
(\ref{uxo}.a,\hskip.7pt c) and (\ref{fso}.a), we now get $\,x_{ttss}=Q(x_s)\hskip.4pt u/2\,$
and $\,\widetilde v_{ss}=0$.

Thus, $\,\widetilde v_s$ must be identically zero, as it is parallel in the
$\,s\,$ direction and vanishes at $\,s=0$. For the same reason,
$\,\widetilde v=0\,$ for all $\,(t,s)$, which yields (\ref{xtt}.i).

By (\ref{xtt}.i), $\,x_{tt}=0\,$ when $\,s=1$, so that the curve
$\,t\mapsto x(t,1)\,$ is a geodesic defined on $\,\text{\bf R}\hskip.7pt$. Such geodesics
realize all initial conditions $\,(x,\dot x)\,$ with the velocities
$\,\dot x\,$ for which $\,g_x(u_x,\dot x)=1\,$ (the normalization being due to
the fact that they are parametrized by the function $\,t$, cf.\
Remark~\ref{param}). Namely, we can realize
$\,(x,\dot x)\,$ by the curve $\,t\mapsto y(t)\,$ chosen above, and then use
the solution $\,w\,$ to (\ref{ode}) with the zero initial conditions.
\end{proof}

\section{Proof of Theorem~\ref{class}}\label{pottc}
Suppose that $\,(\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\hskip.7pt)\,$ satisfies the assumptions of
Theorem~\ref{class}. The data required for the construction in
Section~\ref{xpls} can be introduced as follows. For $\,u\,$ chosen at the
beginning of Section~\ref{clth}, we define $\,f\,$ as in the lines following
(\ref{ruv}). According to (f) in Section~\ref{potb}, $\,f\,$ is, locally, a
function of $\,t$. The word `locally' may be dropped as we are assuming
condition (c) in Lemma~\ref{clext}, and hence the level sets of $\,\hskip.7pt t\hskip.7pt$
are connected. Next, we let $\,V\hskip.7pt$ be the space of all parallel sections
of $\,\mathcal{E}$, so that $\,\dimV\hskip-.7pt=n-2\,$ by Lemma~\ref{csnpd}(iv).
Finally, the pseu\-\hbox{do\hskip.7pt-}Euclid\-e\-an inner product $\,\lg\,,\rg\,$ in
$\,V$ and $\,A:V\toV\hskip.7pt$ are the objects induced, in an
obvious manner, by the fibre metric $\,\gamma\,$ on $\,\mathcal{E}\,$ and the bundle
morphism $\,A:\mathcal{E}\to\mathcal{E}$, both of which are parallel (see Remark~\ref{fibme}
and (d) in Section~\ref{potb}).

We now fix a null geodesic $\,\text{\bf R}\ni t\mapsto x(t)\in\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\,$ parametrized by
the function $\,t$, which exists in view of Lemma~\ref{compl} and
Remark~\ref{param}(a). As $\,g(\dot x,u)=1\,$ (see Remark~\ref{param}(c)), the
plane $\,P\,$ in $\,{T\hskip-2pt_{x(0)}\hskip-.9pt\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt}\,$ spanned by the null vectors $\,\dot x(0)\,$ and
$\,u_{x(0)}$ is $\,g_x$-non\-de\-gen\-er\-ate, with the sign pattern
\hbox{$\,-\hskip1.2pt+\hskip.7pt$}, so that $\,{T\hskip-2pt_{x(0)}\hskip-.9pt\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt}=P\oplus\widetildeV\hskip-.7pt$, for
$\,\widetildeV\hskip-.7pt=P^\perp\hskip-1pt$.

We define a mapping $\,F:\mathbf{R}\hskip-.5pt^2\hskip-.7pt\timesV\hskip-.7pt\to\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt\,$ by
$\,F(t,s,v)=\exp_{\hskip.7pt x(t)}(\tilde v(t)+su_{x(t)}/2)$, for the parallel field
$\,t\mapsto \tilde v(t)\in{T\hskip-2pt_{x(t)}\hskip-.9pt\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt}\,$ with
$\,\tilde v(0)=\hskip.7pt\text{\rm pr}\hskip2.4ptv_{x(0)}$, where
$\,\hskip.7pt\text{\rm pr}:{T\hskip-2pt_{x(0)}\hskip-.9pt\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt}\to\widetildeV\hskip.7pt$ is the orthogonal projection
(and so $\,\hskip.7pt\text{\rm pr}\hskip.4pt$, restricted to $\,\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp$ for $\,x=x(0)$,
descends to the quotient $\,\mathcal{E}_x=\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp\hskip-1pt/\mathcal{D}_x$, forming an isomorphism
$\,\mathcal{E}_x\to\widetildeV$).

That $\,F\,$ is a dif\-feo\-mor\-phism\ can be seen as follows. The manifold $\,N\hskip.7pt$ in
Lemma~\ref{clext}(c) is simply connected, since so is $\,\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt$. Therefore,
each leaf of $\,\dz^\perp$ (level set of $\,t$), with its complete flat
tor\-sion\-free connection (Remark~\ref{fibme}), is the dif\-feo\-mor\-phic\ image
of its tangent space at any point under the exponential mapping,
cf.\ Remark~\ref{scfco}.

Finally, according to \cite[Lemma 5.1]{derdzinski-roter-07}, $\,F^*\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\,$
coincides with the metric $\,\kappa\,dt^2\hskip-.7pt+\hskip.7pt dt\hskip.7pt ds+h$ on
$\,\mathbf{R}\hskip-.5pt^2\hskip-.7pt\timesV\hskip-.7pt$, constructed from the data described above as in
Section~\ref{xpls}.

\section{Proof of Theorem~\ref{nolor}}\label{potc}
In the following lemma, the bundle morphism $\,A:\mathcal{E}\to\mathcal{E}\,$ defined in
Section~\ref{potb} makes sense even without assuming that the line bundle
$\,\mathcal{D}\,$ is trivial. In fact, $\,A\,$ depends quadratically on our fixed
nonzero parallel section $\,u\,$ of $\,\mathcal{D}$, which, for nontrivial $\,\mathcal{D}$,
is still well-defined, locally, up to a sign. (Cf.\ Theorem~\ref{parvf}.)
\begin{lemma}\label{eglfl}Let\/ $\,(M,g)\,$ be a compact essentially con\-for\-mal\-ly symmetric\
Lo\-rentz\-i\-an manifold\ of dimension $\,n\ge4\,$ such that, at some/every
point\/ $\,x\in M$, the nonzero traceless self-ad\-joint endomorphism
$\,A_x:\mathcal{E}_x\to\mathcal{E}_x$ has $\,n-2=\dim\mathcal{E}_x$ distinct eigenvalues. Then the
leaves of the parallel distribution $\,\dz^\perp$ are all complete.
\end{lemma}
In fact, in view of Lemma~\ref{csnpd}(iv) and (e) in Section~\ref{potb},
passing to a suitable finite covering of $\,M\,$ we may assume that both
vector bundles $\,\mathcal{D}\,$ and $\,\mathcal{E}\hskip.7pt$ over $\,M\,$ are trivialized by their
parallel sections. Our assertion now follows from Lemma~\ref{flcpl} applied to
$\,\mathcal{L}\,$ which is the restriction of $\,\mathcal{D}\,$ to
any leaf $\,N\hskip.7pt$ of $\,\dz^\perp\hskip-1pt$.
\begin{remark}\label{eigvl}For $\,n=4$, the assumption about eigenvalues in
Lemma~\ref{eglfl} is redundant: it follows from the other stated properties of
$\,A_x$, since $\,\gamma_x$ is positive definite (Remark~\ref{fibme}).
\end{remark}
\begin{proof}[Proof of Theorem~{\rm\ref{nolor}}]Suppose that $\,(M,g)\,$
is a compact essentially con\-for\-mal\-ly symmetric\ Lo\-rentz\-i\-an four-man\-i\-fold. Its universal covering
$\,(\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\hskip.7pt)\,$ then satisfies the assumptions of Theorem~\ref{class}: the
leaves of $\,\dz^\perp$ are complete by Lemma~\ref{eglfl} (cf.\ Remark~\ref{eigvl}),
while condition (c) in Lemma~\ref{clext} holds for $\,t\,$ in view of
Remark~\ref{concl}.

The conclusion of Theorem~\ref{class} now contradicts
Lemma~\ref{ecsrr}(iv-b).
\end{proof}

\section{Vector bundles related to Killing fields}\label{vbkf}
Throughout this section, $\,(M,g)\,$ stands for a fixed essentially con\-for\-mal\-ly symmetric\ pseu\-\hbox{do\hs-}Riem\-ann\-i\-an manifold\ of
dimension $\,n\ge4\,$ such that the Ol\-szak distribution\ $\hskip.7pt\mathcal{D}\hskip.7pt$ is two-di\-men\-sion\-al, and
$\,\hskip.7pt\omega\hskip.7pt\,$ is the $\,2$-form described in Remark~\ref{oldis}. We assume
that $\,\hskip.7pt\omega\hskip.7pt\,$ is single-val\-ued, rather than being defined just up to
a sign, which can always be achieved by passing to a two-fold covering
manifold, if necessary.

In addition to $\,\mathcal{D}\,$ and $\,\dz^\perp\hskip-1pt$, we consider here the real vector
bundle $\,\mathcal{Y}\,$ over $\,M$, the sections of which are the differential
$\,2$-forms $\,\zeta\,$ such that $\,\zeta(v,\,\cdot\,)=0\,$ for every section
$\,v\,$ of $\,\mathcal{D}$. Thus, $\,\mathcal{Y}\,$ is a subbundle of
$\,({T^*\!M})^{\wedge2}\hskip-1pt$, isomorphic to $\,([\hskip.4pt{T\hskip-.3ptM}/\mathcal{D}]^*)^{\wedge2}\hskip-1pt$,
and $\,\hskip.7pt\omega\hskip.7pt\,$ is a section of $\,\mathcal{Y}\,$ (as
$\,\mathcal{D}\subset\hskip.7pt\text{\rm Ker}\,\hskip.7pt\omega\hskip.7pt\,$ by (\ref{krp}.a) and (\ref{owo}).
Let $\,\mathcal{L}\,$ be the real-line subbundle of $\,\mathcal{Y}$ spanned by
$\,\hskip.7pt\omega\hskip.4pt$. The Le\-vi-Ci\-vi\-ta connection\ of $\,g\,$ induces connections, all denoted by
$\,\nabla$, in each of the byndles $\,\mathcal{D},\dz^\perp\hskip-1pt,\mathcal{Y}\,$ and $\,\mathcal{L}$.

The formula
$\,\hbox{$\nabla\!\!^{^{^{_{\!\!\circ}}}}$}_{\!u}(v,\zeta)=(\nabla_{\!u}v-\zeta u,\nabla_{\!u}\zeta
-(n-2)^{-1}g(v,\,\cdot\,)\wedge{\rho}\hskip.4pt(u,\,\cdot\,))$, with $\,{\rho}\,$ standing
for the Ricci tensor\ of $\,g$, clearly defines a connection $\,\hbox{$\nabla\!\!^{^{^{_{\!\!\circ}}}}$}\,$ in the vector bundle\
$\,\dz^\perp\hskip-1pt\oplus\mathcal{Y}$. (Here $\,\zeta u\,$ is the unique vector field with
$\,g(\zeta u,\,\cdot\,)=\zeta(u,\,\cdot\,)$, and similarly for the symbols
$\,{\rho} u,\hskip.7pt\omega_x$ appearing below.) The following facts will be needed in
the next section.
\begin{enumerate}
  \def{\rm\alph{enumi}}{{\rm\alph{enumi}}}
\item The connection $\,\hbox{$\nabla\!\!^{^{^{_{\!\!\circ}}}}$}\,$ in $\,\dz^\perp\hskip-1pt\oplus\mathcal{Y}\,$ is flat.
\item The subbundle\ $\,\mathcal{D}\oplus\mathcal{L}\,$ of $\,\dz^\perp\hskip-1pt\oplus\mathcal{Y}\,$ is
$\,\hbox{$\nabla\!\!^{^{^{_{\!\!\circ}}}}$}$-par\-al\-lel.
\item For every $\,\hbox{$\nabla\!\!^{^{^{_{\!\!\circ}}}}$}$-par\-al\-lel\ section $\,(v,\zeta)\,$ of $\,\dz^\perp\hskip-1pt\oplus\mathcal{Y}$,
the $\,g$-co\-var\-i\-ant derivative $\,\nabla v\,$ is $\,\nabla$-par\-al\-lel\ along
$\,\dz^\perp\hskip-1pt$.
\end{enumerate}
If $\,M\,$ is also simply connected, then, for the space $\,\mathcal{V}\,$ of all
$\,\hbox{$\nabla\!\!^{^{^{_{\!\!\circ}}}}$}$-par\-al\-lel\ sections of $\,\mathcal{D}\oplus\mathcal{L}$,
\begin{enumerate}
  \def{\rm\alph{enumi}}{{\rm\alph{enumi}}}
\item[(d)] $\dim\mathcal{V}=3$,
\item[(e)] there exists a unique $\,C^\infty$ mapping
$\,F:M\to\mathcal{V}\smallsetminus\hskip-.7pt\{0\}\,$ such that $\,F(x)$, for $\,x\in M$, is
characterized by $\,F(x)=(v,\zeta)\,$ with $\,v_x=0\,$ and $\,\zeta_x=\hskip.7pt\omega_x$,
\item[(f)] for $\,F\,$ as in (e) and any $\,x\in M$, the differential
$\,dF_x:{T\hskip-2pt_x\hskip-.9ptM}\to\mathcal{V}\hskip.7pt$ has rank $\,2$, while the image $\,dF_x({T\hskip-2pt_x\hskip-.9ptM})\,$ and
$\,F(x)\,$ together span $\,\mathcal{V}$,
\item[(g)] the leaves of $\,\dz^\perp$ are the connected components of nonempty
$\,F$-pre\-im\-ages of points of $\,\mathcal{V}$,
\item[(h)] for every leaf $\,N\hskip.7pt$ of $\,\dz^\perp\hskip-1pt$, the tangent bundle
$\,{T\hskip-.3ptN}\hskip.7pt$ is trivialized by its $\,\nabla$-par\-al\-lel\ sections (cf.\
Remark~\ref{fibme}).
\end{enumerate}
We will not use the easily-verified fact that the assignment
$\,v\mapsto(v,\zeta)$, with $\,\zeta\,$ characterized by
$\,\zeta(u,\,\cdot\,)=g(\nabla_{\!u}v,\,\cdot\,)\,$ for all vector fields
$\,u$, is a linear isomorphism between the space of all Killing field s $\,v\,$ on
$\,(M,g)\,$ tangent to $\,\dz^\perp\hskip-1pt$, and the space of all $\,\hbox{$\nabla\!\!^{^{^{_{\!\!\circ}}}}$}$-par\-al\-lel\
sections $\,(v,\zeta)\,$ of $\,\mathcal{Y}$.

Let $\,\hbox{$R\hskip-4.5pt^{^{^{_{\circ}}}}$}{},R\,$ and $\,\hat R\,$ be the curvature tensors of $\,\hbox{$\nabla\!\!^{^{^{_{\!\!\circ}}}}$}$, the
Le\-vi-Ci\-vi\-ta connection\ $\,\nabla\,$ of $\,g\,$ and, respectively, the connection induced by
$\,\nabla\,$ in $\,[{T^*\!M}]^{\wedge2}\hskip-1pt$. For arbitrary vector fields
$\,u,u\hskip.4pt'$ on $\,M$, (\ref{cur}) yields
$\,\hbox{$R\hskip-4.5pt^{^{^{_{\circ}}}}$}{}(u,u\hskip.4pt'\hskip.7pt)\hskip.4pt(v,\zeta)=(v\hskip.4pt'\hskip-.7pt,\zeta\hskip.7pt'\hskip.7pt)$, where
$\,v\hskip.4pt'\hskip-.7pt=R(u,u\hskip.4pt'\hskip.7pt)\hskip.4pt v
-\text{\hbox{$(n-2)^{-1}g(v,u){\rho} u\hskip.4pt'\hskip-.7pt$}}
+\text{\hbox{$(n-2)^{-1}g(v,u\hskip.4pt'\hskip.7pt){\rho} u$}}\,$ and
$\,\zeta\hskip.7pt'\hskip-.7pt=\hat R(u,u\hskip.4pt'\hskip.7pt)\hskip.4pt\zeta
+(n-2)^{-1}[\hskip.7pt\zeta(u,\,\cdot\,)\wedge{\rho}(u\hskip.4pt'\hskip-1pt,\,\cdot\,)
-\zeta(u\hskip.4pt'\hskip-1pt,\,\cdot\,)\wedge{\rho}(u,\,\cdot\,)\hskip.4pt]$. (By
Lemma~\ref{rwcod}(b), $\,(\nabla_{\!u}{\rho})(u\hskip.4pt'\hskip-1pt,\,\cdot\,)\,$ is
symmetric in $\,u,u\hskip.4pt'\hskip-1pt$.) Now Lemma~\ref{rwcod}(a) gives
$\,g(v\hskip.4pt'\hskip-1pt,\,\cdot\,)=W(u,u\hskip.4pt'\hskip-1pt,v,\,\cdot\,)$. (Note that
$\,{\rho}(v,\,\cdot\,)=0\,$ in view of (\ref{krp}.a), since $\,v\,$ is a section
of $\,\dz^\perp\hskip-1pt$.) Consequently, $\,v\hskip.4pt'\hskip-.7pt=0$, as $\,v\,$ is a section of
$\,\dz^\perp\hskip-.7pt=\hskip.7pt\text{\rm Ker}\,\hskip.7pt\omega\hskip.7pt\subset\hskip.7pt\text{\rm Ker}\hskip2ptW\hskip-.7pt$, by (\ref{owo}) with
$\,W\hskip-.7pt=\hskip.7pt\varepsilon\hskip1pt\omega\hskip-.7pt\otimes\omega\hskip.4pt$.
Furthermore, by the Ric\-ci identity,
$\,[\hskip.4pt\hat R(u,u\hskip.4pt'\hskip.7pt)\hskip.4pt\zeta\hskip.7pt](w,w\hskip.4pt'\hskip.7pt)
=\zeta(R(u\hskip.4pt'\hskip-1pt,u)\hskip.4pt w,w\hskip.4pt'\hskip.7pt)+\zeta(w,R(u\hskip.4pt'\hskip-1pt,u)\hskip.4pt w\hskip.4pt'\hskip.7pt)$
for any vector fields $\,w,w\hskip.4pt'\hskip-1pt$. Replacing $\,R\,$ here by the
expression in Lemma~\ref{rwcod}(a), we get $\,\zeta\hskip.7pt'\hskip-.7pt=0$. (In fact,
numerous terms vanish since the images of $\,W_{\hskip-.7pt x}$ and $\,{\rho}_x$ at
any point $\,x\,$ are contained in
$\,\dz\hskip-.3pt_x\subset\hskip.7pt\text{\rm Ker}\,\zeta_x$, cf.\ (\ref{owo}) and
(\ref{krp}.a).) This proves (a).

Next, if $\,(v,\zeta)\,$ is a section of $\,\mathcal{D}\oplus\mathcal{L}$, then, for any vector
field $\,u$, so are $\,(\nabla_{\!u}v,\nabla_{\!u}\zeta)$ (since $\,\mathcal{D}\,$ and
$\,\hskip.7pt\omega\hskip.7pt\,$ are parallel), and
$\,(\zeta u,(n-2)^{-1}g(v,\,\cdot\,)\wedge{\rho}\hskip.4pt(u,\,\cdot\,)\,$ (as $\,\mathcal{D}\,$
is the image of $\,\hskip.7pt\omega\hskip.7pt\,$ by (\ref{owo}), and $\,\zeta\,$ equals a
function times $\,\hskip.7pt\omega\hskip.4pt$, while $\,v\,$ and $\,{\rho} u\,$ are sections of
$\,\mathcal{D}$, cf.\ Lemma~\ref{csnpd}(ii)). Consequently, $\,\hbox{$\nabla\!\!^{^{^{_{\!\!\circ}}}}$}_{\!u}(v,\zeta)\,$ is
a section of $\,\mathcal{D}\oplus\mathcal{L}\,$ as well, and (b) follows.

By the definition of $\,\hbox{$\nabla\!\!^{^{^{_{\!\!\circ}}}}$}$, if $\,(v,\zeta)\,$ is a $\,\hbox{$\nabla\!\!^{^{^{_{\!\!\circ}}}}$}$-par\-al\-lel\ section of
$\,\dz^\perp\hskip-1pt\oplus\mathcal{Y}$, then $\,\nabla_{\!u}\zeta=0\,$ for every section $\,u\,$
of $\,\dz^\perp\hskip-1pt$, as one then has $\,{\rho}\hskip.4pt(u,\,\cdot\,)=0\,$ by (\ref{krp}.a).
Hence $\,\zeta\,$ is $\,\nabla$-par\-al\-lel\ along $\,\dz^\perp\hskip-1pt$, which yields (c).

Now let $\,M\,$ be simply connected. Assertion (d) is obvious from (a) and
(b), as $\,3\,$ is the fibre dimension of $\,\mathcal{D}\oplus\mathcal{L}$. Similarly, (e)
is due to the existence of a unique parallel section of $\,\mathcal{D}\oplus\mathcal{L}\,$
with a prescribed value at $\,x$.

To prove (f), we first observe that $\,dF_x$ sends any $\,u\in{T\hskip-2pt_x\hskip-.9ptM}\,$ to
$\,(\dot v,\dot\zeta)\in\mathcal{V}\,$ characterized by
$\,\dot v_x=-\hskip.7pt\omega_xu\,$ and $\,\dot\zeta_x=0$. In fact, let
$\,t\mapsto x(t)\,$ be a curve in $\,M$, and let us set
$\,(v(t),\zeta(t))=F(x(t))$. Suppressing the dependence on $\,t$, and
differentiating, covariantly along the curve, both the relation $\,v_x\hskip-.7pt=0\,$
and the equality which states that $\,(\nabla v)_x$ corresponds via $\,g_x$ to
$\,\hskip.7pt\omega_x$, we get $\,\dot v_x+\hskip.7pt\omega_x(\dot x,\,\cdot\,)=0\,$ and
$\,(\nabla\dot v)_x=0$, as required. (The second covariant derivative of the
Killing field\ $\,v\,$ at $\,x\,$ depends linearly on $\,v_x$, due to a well-known
identity \cite[formula (17.4) on p.\ 536]{dillen-verstraelen}, and so
$\,\nabla(\nabla v)=0\,$ at $\,x$, since $\,v_x=0$.) As
$\,\hskip.7pt\text{\rm rank}\,\hskip.7pt\omega\hskip.7pt=2\,$ (see Remark~\ref{oldis}), this
implies (f).

As a consequence of (c), $\,F\,$ is constant along $\,\dz^\perp\hskip-1pt$. (Note that
$\,\nabla\hskip.7pt\omega\hskip.7pt=0\,$ and, by (\ref{owo}),
$\,\dz^\perp\hskip-.7pt=\hskip.7pt\text{\rm Ker}\,\hskip.7pt\omega\hskip.4pt$.) Now (g) is immediate from (f).

Finally, in view of (a), given a leaf $\,N\hskip.7pt$ of $\,\dz^\perp\hskip-1pt$, a point
$\,x\in N\hskip-.7pt$, and a vector $\,w\in{T\hskip-2pt_x\hskip-.9ptN}=\dz\hskip-.3pt_x{}\hskip-4.2pt^\perp\hskip-1pt$, we may choose a
$\,\hbox{$\nabla\!\!^{^{^{_{\!\!\circ}}}}$}$-par\-al\-lel\ section $\,(v,\zeta)\,$ of $\,\dz^\perp\hskip-1pt\oplus\mathcal{Y}\,$ satisfying the
initial conditions $\,v_x=w\,$ and $\,\zeta_x=0\,$ (that is,
$\,(\nabla v)_x=0$). Thus, $\,v\,$ is tangent to $\,N\hskip.7pt$ and, by (c),
$\,\nabla$-par\-al\-lel\ along $\,N\hskip-.7pt$, which proves (h).

\section{Proof of the second part of Theorem~\ref{pontr}}\label{psta}
We need the following two simple facts from topology.
\begin{lemma}\label{abfgp}If the fundamental group\/ $\,\Gamma$ of a
compact\/ $\,k$-di\-men\-sion\-al\ manifold\/ $\,P$ is Abelian and the universal
covering manifold\ of\/ $\,P\hskip-.7pt\,$ is diffeomorphic to $\,{\mathbf{R}}^{\hskip-.6ptk}\hskip-.7pt$, then\/
$\,\Gamma$ is isomorphic to $\,\text{\bf Z}^k\hskip-.7pt$.
\end{lemma}
\begin{proof}As $\,\Gamma\hskip.4pt$ is tor\-sion\-free by Smith's theorem
\cite[p.\ 287]{hu}, and finitely generated, it is isomorphic to
$\,\text{\bf Z}^{\hskip.7pt r}$ for some integer $\,r\ge1$. The $\,K(\text{\bf Z}^{\hskip.7pt r}\hskip-.7pt,1)\,$
space $\,P\,$ must have the homotopy type of the $\,r$-to\-rus
\cite[pp.\ 93--95]{switzer}, so that $\,r=k$, since both $\,r\,$ and $\,k\,$
are equal to the highest integer $\,m\,$ with $\,H_m(P,\text{\bf Z}_2)\ne\{0\}$.
\end{proof}
\begin{lemma}\label{infgp}If\/ $\,M\to S^2$ is a fibration and its fibre
$\,N\hskip.7pt$ is a compact manifold of dimension $\,k\ge2\,$ with a universal
covering space dif\-feo\-mor\-phic\ to $\,{\mathbf{R}}^{\hskip-.6ptk}\hskip-1pt$, then the fundamental group of\/
$\,M\,$ is infinite.
\end{lemma}
\begin{proof}Since $\,\text{\bf Z}=\pi_2S^2\to\hskip.7pt\pi_1N\to\pi_1M\,$ is a part of
the homotopy exact sequence of the fibration $\,M\to S^2\hskip-1pt$, if
$\,\pi_1M\,$ were finite, the image $\,\Gamma\hskip.7pt$ of $\,\text{\bf Z}=\pi_2S^2$ would
be a cyclic subgroup of finite index in $\,\pi_1N\hskip-.7pt$. The manifold
$\,P={\mathbf{R}}^{\hskip-.6ptk}\hskip-.7pt/\Gamma$, forming a finite covering space of
$\,N\hskip-.7pt={\mathbf{R}}^{\hskip-.6ptk}/\pi_1N\hskip-.7pt$, would be compact, which, as $\,k\ge2$, would
contradict Lemma~\ref{abfgp}.
\end{proof}
We now assume that $\,(M,g)\,$ is a compact simply connected essentially con\-for\-mal\-ly symmetric\ manifold. As we
show below, this assumption leads to a contradiction, which proves the claim
about $\,\pi_1M\,$ in Theorem~\ref{pontr}.

Let $\,d\in\{1,2\}\,$ be the dimension of the Ol\-szak distribution\ $\,\mathcal{D}\,$ (see
Lemma~\ref{csnpd}(i)).

If $\,d=1$, Lemma~\ref{csnpd}(iv) implies the existence of a nonzero
global parallel vector field $\,u\,$ spanning $\,\mathcal{D}$. The $\,1$-form
$\,\xi=g(u,\,\cdot\,)$, being parallel, is closed, so that $\,\xi=dt\,$
for some function $\,t$. As $\,dt\,$ is parallel, $\,dt\ne0\,$
everywhere, which contradicts compactness of $\,M$.

Now let $\,d=2$, and let $\,\mathcal{V},F\,$ be as in (d) -- (g),
Section~\ref{vbkf}. The formula $\,\pi(x)=F(x)/|F(x)|\,$ defines a submersion
$\,\pi:M\to S^2\hskip-1pt$, where $\,S^2\hskip-.7pt=\{w\in\mathcal{V}:|w|=1\}\,$ is the unit sphere
for a fixed Euclidean norm $\,|\hskip2.4pt|\,$ in the $\,3$-space $\,\mathcal{V}$. As
$\,\pi(M)\subset S^2$ is both compact and open, the submersion $\,\pi\,$ is
surjective, so that $\,\pi\,$ is a fibration (Remark~\ref{fibra}). The fibres of
$\,\pi\,$ thus are the leaves of $\,\dz^\perp$ (see (g) in Section~\ref{vbkf}). If
a fixed fibre $\,N\hskip.7pt$ is endowed with the flat tor\-sion\-free connection
mentioned in Remark~\ref{fibme}, then, according to (h) in Section~\ref{vbkf},
$\,{T\hskip-.3ptN}\hskip.7pt$ is trivialized by its parallel sections. Lemma~\ref{flcpl} (for
$\,\mathcal{L}={T\hskip-.3ptN}$) and Remark~\ref{scfco} now imply that the universal covering
manifold of $\,N\hskip.7pt$ is dif\-feo\-mor\-phic\ to $\,\text{\bf R}^{\hskip-.6ptn-2}\hskip-1pt$. Since
$\,\pi_1M\,$ was assumed to be trivial, and $\,n-2\ge2$, this contradicts
Lemma~\ref{infgp}, thus completing the proof of Theorem~\ref{pontr}.

\section{Further remarks}\label{fure}
This section consists of three separate comments, indicating how some results
presented above might be strengthened.

First, Theorem~\ref{class}, with essentially the same proof, remains valid if,
in its assumptions, condition (c) in Lemma~\ref{clext} and completeness of the
leaves of $\,\dz^\perp$ are replaced by completeness of $\,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt$.

Secondly, the argument that we used to show nonexistence of compact
four-di\-men\-sion\-al\ essentially con\-for\-mal\-ly symmetric\ Lo\-rentz\-i\-an manifold s can be minimally modified so as to
yield the following classification theorem: {\it If a compact essentially con\-for\-mal\-ly symmetric\
Lo\-rentz\-i\-an manifold\/ $\,(M,g)\,$ of any dimension $\,n\ge4\,$ satisfies the
assumption about distinct eigenvalues made in Lemma\/ {\rm\ref{eglfl}}, then
the pseu\-\hbox{do\hs-}Riem\-ann\-i\-an\ universal covering $\,(\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\hskip.7pt)\,$ of\/ $\,(M,g)\,$ coincides, up
to an isometry, with one of the manifolds constructed in
Section\/~{\rm\ref{xpls}}, and the fundamental group of\/ $\,M$, treated as a
group of isometries of $\,(\hskip1.9pt\widehat{\hskip-1.9ptM\hskip-.2pt}\hskip.2pt,\hskip1.2pt\widehat{\hskip-1.2ptg\hskip-.4pt}\hskip.4pt\hskip.7pt)$, has a fi\-nite-index subgroup
contained in the group $\,\mathrm{G}\hskip.7pt$ defined in Section\/~{\rm\ref{xpls}}}.

Finally, in Section~\ref{psta} we showed that $\,\pi_1M\,$ is infinite for any
compact essentially con\-for\-mal\-ly symmetric\ manifold\ $\,(M,g)$, using separate arguments for the cases
$\,d=1\,$ and $\,d=2$, where $\,d\,$ is the dimension of the Ol\-szak distribution\
$\,\mathcal{D}$. If $\,d=1$, we obtain the stronger conclusion $\,b_1(M)\ge1\,$ from
the following lemma applied to $\,\mathcal{F}=\dz^\perp$ along with a
vec\-tor-bun\-dle isomorphism $\,\mathcal{D}\to({T\hskip-.3ptM}/\mathcal{F})^*$ provided by $\,g$,
cf.\ Lemma~\ref{csnpd}(iv).
\begin{lemma}\label{betti}Let a compact manifold\/ $\,M\hskip.7pt$ with a
tor\-sion\-free connection $\,\nabla\hskip.7pt$ admit a co\-di\-men\-sion-one
$\,\nabla$-par\-al\-lel\ distribution\/ $\,\mathcal{F}$ such that the
connection induced by $\,\nabla\hskip.7pt$ in the quotient line bundle\/
$\,\hskip.7pt{T\hskip-.3ptM}/\mathcal{F}\,$ is flat. Then the first Betti number\/ $\,b_1(M)\hskip.7pt$
is positive, and, consequently, $\,M\hskip.7pt$ has an infinite fundamental group.
\end{lemma}
\begin{proof}The dual bundle of $\,\hskip.7pt{T\hskip-.3ptM}/\mathcal{F}\hskip.7pt$ may be
identified with the real-line subbundle $\,\mathcal{L}$ of $\,{T^*\!M}\,$
such that the sections of $\,\mathcal{L}\,$ are precisely those $\,1$-forms
$\,\xi\,$ on $\,M\,$ with $\,\xi(w)=0$ for every section $\,w\,$ of
$\,\mathcal{F}$. Clearly, $\,\mathcal{L}\,$ is $\,\nabla$-par\-al\-lel.

Suppose, on the contrary, that $\,b_1(M)=0$, so that the homology
group $\,H_1(M,\text{\bf Z})\,$ is finite. Replacing $\,M\,$ by a two-fold covering
manifold, if necessary, we may assume that $\,\mathcal{L}\,$ is spanned by a global nonzero
parallel $\,1$-form $\,\xi$. In fact, the connection in $\,\mathcal{L}\,$ induced by
$\,\nabla\,$ is flat, and so its holonomy representation, with any fixed
base point $\,x\in M$, its valued in the multiplicative group
$\,\text{\bf R}\smallsetminus\{0\}$. Since $\,\text{\bf R}\smallsetminus\{0\}\,$ is Abelian,
the holonomy representation is a composite
$\,\pi_1M\to H_1(M,\text{\bf Z})\to\text{\bf R}\smallsetminus\{0\}$, and its image must, due
to finiteness of $\,H_1(M,\text{\bf Z})$, be contained in $\,\{1,-1\}$.

As $\,\nabla\,$ is tor\-sion\-free and $\,b_1(M)=0$, the parallel
$\,1$-form $\,\xi\,$ is closed, and hence $\,\xi=dt$ is nonzero
everywhere, for some function $\,t$, which contradicts compactness of $\,M$.
\end{proof}

redpajama_set_name: RedPajamaArXiv


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text:
Q: How can I change the Visual Studio default Form font? I had some problems with my Windows fonts, because of that, VS changed the default Form font to Arial. How can I go back to Microsoft Sans Serif ?
I know that I can change the Font at the Form propertie, but I would have to change it at all forms that I create.
Edit:
I found out how I've messed up with the windows Fonts, I deleted all entries of the Win.ini file because my filepath was null and "If this parameter is NULL, the function searches the Win.ini file."
[DllImport("KERNEL32.DLL", EntryPoint = "GetPrivateProfileSectionNamesA", CharSet = CharSet.Ansi)]
private static extern int GetPrivateProfileSectionNames(byte[] lpszReturnBuffer, int nSize, string lpFileName);

As stuard said, VS gets the font from DEFAULT_GUI_FONT. I still don't know how to reset it back to Microsoft Sans Serif.

A: It may be Windows which is at fault rather than Visual Studio, as according to this page the forms query Windows for DEFAULT_GUI_FONT.

redpajama_set_name: RedPajamaStackExchange


text:
Q: NSURLConnection different result when get error I'm using -[NSURLConnection initWithRequest...] to connect to a server,
and when the server side fails it will respond with an error 400 with a message in the body.
When there is no error, I can get response data in didReceiveData. But when the server returns an error 400, I do not receive any error message in didReceiveData.
However if I use -[NSURLConnection sendSynchronousRequest...] to do the same request
I do get error data from sendSynchronousRequest's return value.
I want to get the error message from didReceiveData, but I dont know why I cannot get it.
What is different between the do, can someone help me?
This way I can get the error message:
NSURL *url = [NSURL URLWithString:@"http://devapi.xxx.com/v1/debug/error"];
NSMutableURLRequest *request = [[[NSMutableURLRequest alloc] initWithURL:url
                                                              cachePolicy:NSURLRequestUseProtocolCachePolicy
                                                          timeoutInterval:240] autorelease];

[request setHTTPMethod:@"POST"];

[request setValue:@"application/json" forHTTPHeaderField:@"Accept"];
[request setValue:@"application/json" forHTTPHeaderField:@"Content-Type"];

NSURLResponse *response = nil;
NSError *error = nil;
NSData *data = [NSURLConnection sendSynchronousRequest:request
                                     returningResponse:&response
                                                 error:&error];

NSLog(@"STATUS:%d", [((NSHTTPURLResponse *)response) statusCode]);
NSLog(@"ERROR:%@", error);

This way I cannot get it:
NSMutableURLRequest *request = [[[NSMutableURLRequest alloc] initWithURL:[NSURL URLWithString:@"http://devapi.xxx.com/v1/debug/error"]
                                                             cachePolicy:NSURLRequestUseProtocolCachePolicy
                                                         timeoutInterval:240] autorelease];

[request setHTTPMethod:@"POST"];

[request setValue:@"application/json" forHTTPHeaderField:@"Accept"];
[request setValue:@"application/json" forHTTPHeaderField:@"Content-Type"];
urlConnection=[[NSURLConnection alloc] initWithRequest:request delegate:self startImmediately:YES];

- (void)connection:(NSURLConnection *)connection didReceiveResponse:(NSURLResponse *)response {
    NSLog(@"statusCode:%d",((NSHTTPURLResponse *)response).statusCode);
}

- (void)connection:(NSURLConnection *)connection didFailWithError:(NSError *)error {
}
- (void)connection:(NSURLConnection *)connection didReceiveData:(NSData *)data {
[receivedData appendData:data];
NSLog(@"DATA:%@",[NSString stringWithCString:[receivedData bytes] encoding:NSASCIIStringEncoding]);
}

A: Use this delegate method for handling such errors.
- (void)connection:(NSURLConnection *)connection didFailWithError:(NSError *)error {
      NSLog(@" Failed with error---%@",error);
}

A: I think I found what happen it is,
When I get a error status code in didReceiveResponse,
I immediately try to get data and cancel connection,
But it is wrong,
Because didReceiveData still in receive after didReceiveResponse.
Thanks a lot!


text:
Q: Can someone help debug this Select Case I've been at this for a little while, with no succes.
It won't show any other result, except for the first two cases and the last one.
I have tried an less than or equal to case.
        Dim DayLeft As Long
        DaysLeft = Me.Days.Value

        Select Case DaysLeft
            Case Is = 0
                MsgBox "Deadline due today!"
            Case Is < 0
                MsgBox "Deadline expired!"
            Case 90 To 31
                MsgBox "First notice " & Me.Days& " days until deadline!"
            Case 30 To 11
                MsgBox "Deadline due soon! " & Me.Days & " days until deadline!"
            Case Is <= 10
                MsgBox "Warning! " & Me.Days& " days until deadline!"
        End Select

A: From the Select Case documentation:

The To keyword specifies a range of values. If you use the To keyword, the smaller value must appear before To.

So you need
Case 31 to 90
...
Case 11 to 30

A: These two are illegal:

    Case 90 To 31
        MsgBox "First notice " & Me.Days& " days until deadline!"
    '...
    Case Is <= 10
        MsgBox "Warning! " & Me.Days& " days until deadline!"

The & operator in Me.Days& gets interpreted as a type hint, not a string concatenation operator: it needs a leading space!
        Case 90 To 31
            MsgBox "First notice " & Me.Days & " days until deadline!"
        '...
        Case Is <= 10
            MsgBox "Warning! " & Me.Days & " days until deadline!"

Without that space, your code won't compile ("Syntax Error").
Rubberduck picks up two unreachable Case clauses, which are unreachable for the reason highlighted in BigBen's answer.

Note that it also warns about DaysLeft being undeclared, and Option Explicit not being specified.


Psychiatrists say that a childs values are pretty much locked in by the age of ten. I grew up a child of suburban Chicago in the mid 1980?s. I had a nice, cold, and snowy day for my birthday since I was born on December 30, 1985 and December is the peak of winter. My values and beliefs will be very different from someone that grew up in Coldwater, Michigan. Different religions also affect this lot since my Hindu beliefs and ideals would be entirely different from a Christian?s beliefs and ideals even if we grew up in the same neighborhood.
Family plays a large part in my life, but it has not usually had a good impact upon it. Throughout my existence, the greed of other family members has plagued me with money problems. It would appear that, to them, having material possessions more important than assuring my well-being.