ON FOUR-DIMENSIONAL CONFORMAL STRUCTURES

M. A. Akivis

Key words: Conformal structure, isotropic fiber bundle,
tensor of conformal curvature, principal distributions,
completely isotropic submanifolds, Petrov's classification.

1991 Math. Subject Classification: 53A30.

Abstract: The conformal structures $CO (4, 0), CO (1, 3)$
and $CO (2, 2)$ are studied on a real manifold $M, \;
\dim M = 4$. On $M$  isotropic fiber bundles $E_\alpha$ and
$E_\beta$ are constructed. These bundles are real for the
$CO (2, 2)$-structure, and they satisfy the condition
 $\overline{E}_\alpha
= E_\beta$ for the $CO (1, 3)$-structure, and  the conditions
$\overline{E}_\alpha = E_\alpha, \;
\overline{E}_\beta = E_\beta$ for the \mbox{$CO (4)$-structure}.
The tensor $C$ of conformal curvature splits  into
two subtensors $C_\alpha$ and $C_\beta$ which are the
curvature tensors of the  bundles $E_\alpha$ and
$E_\beta$, respectively. These subtensors satisfy the same
conditions as the bundles $E_\alpha$ and $E_\beta$.
Conformally semiflat and flat structures and their
geometrical characteristics are studied.
The principal 2-directions are defined,
and conditions for their integrability are obtained.
These investigations for the $CO (1, 3)$-structure are connected
with  Petrov's classification of Einstein's spaces.
Although the paper is of a servey nature, it also contains some
new results.