LORENTZIAN MANIFOLD WITH A GROUP OF CONFORMAL TRANSFORMATIONS
THAT HAS A NORMAL ONE-PARAMETER HOMOTHETY SUBGROUP

M. Podoksenov

1991 Math. Subject Classification: 53C50.

Key words: Conformal transformations of Lorentzian manifold.

From the paper:

\proclaim
{Theorem. (C.Barbance)} Let $(M,g)$ be a complete homogeneous Lorentzian
manifold and $G$ be a group of conformal transformations, which acts on $M$.
Suppose, that for some $p\in G$ a stable subgroup $G_p$ contains a
connected homothety subgroup $H$, which is a normal subgroup in $G$.
Then $G$ is a homothety group.
\endproclaim

This result was proved by the author [2] in a more general case.
We declined the assumptions of homogeneity for $(M,g)$ and of connectedness
for $H$ and proved that $(M,g)$ is a plain-wave space; the actions of
$G$ and $H$ on $M$ were described. Then we found the necessary and
sufficient condition of homogeneity of such manifold and finally proved the
theorem using the global technique. Instead of completeness assumption for $M$,
some restrictions on its metric were used; only the conformally flat manifolds
from the very limited class does not satisfy this restrictions.

Now we decline the assumption, that homothety group $\Ph $ has a stable point,
but we consider only the case, when $\Ph$ is a one-parameter group.