## Split signature 4-dimensional metrics, real totally null planes and
(2,3,5) distributions

### Pawel Nurowski

(Warsaw)

**Abstract:**
This is a report on a joint work with Daniel An.

On a natural circle bundle *T(M)* over a *4*-dimensional manifold
*M* equipped with a split signature metric *g*, whose fibers are real
totally null *2*-planes of a given selfduality, we consider a
tautological rank *2* distribution *D* obtained by lifting each
totally null plane horizontally to its point in the fiber. If the metric *g* is
not antiselfdual the distribution *D* is *(2,3,5)* in *T(M)*, except at most
*4* points in the fiber, which correspond to the algebraic degeneracy of
the selfdual Weyl tensor. We argue, that if *M* is a Cartesian product
of two Riemann surfaces *(S*_{1},g_{1}) and *(S*_{2},g_{2}), and if *g=g*_{1}-g_{2},
then the circle bundle *T(S*_{1} x S_{2}) is just the configuration space for
the physical system of two solid bodies *B*_{1} and *B*_{2}, bounded by the
surfaces *S*_{1} and *S*_{2} and rolling on each other. The condition for the
two bodies to roll on each other without slipping and twisting,
identifies the restricted velocity space for such a system, with the
tautological distribution *D* on *T(S*_{1} x S_{2}). Among the questions we
address is the following one: which bodies can roll on each other
without slipping and twisting in such a way that their restricted
velocity space, identified with the distribution *D* on *T(S*_{1} x S_{2}) has
simple Lie group *G2* as a group of its symmetries. Apart from the well
known situation when the boundaries *S*_{1} and *S*_{2} of the two bodies have
constant curvatures whose ratio is *1:9*, we unexpectedly find a
*1*-parameter family of surfaces *S*_{1} which when bounding a body rolling
without slipping and twisting on a plane have *D* with the symmetry
group *G2*. Although we've found the differential equations for the
curvatures of *S*_{1} and *S*_{2} to define *D* with *G2* symmetry, we are unable
to solve them in full generality so far.
We also consider the bundle *T(M)* and the corresponding distribution *D*
for more general *4*-manifolds then just a product of two surfaces. In
the case of *M* being equipped with the Plebanski 2nd heavenly metric
(not neccessarilly satisfying Plebanski equation), we are able to show
that necessary condition for the symmetry of the distribution *D* on
*T(M)* to be *G2* is that the first derivative of the Plebanski's function
hast to satisfy a certain 7th order ODE with remarkable properties.
Under further assumption of high symmetries for the Plebanski function
this equation is not only necessary but also sufficient for the
symmetry *G2* of *D*.