L3

Cover a plane with curves, one curve through each point

in each direction. How can you tell whether these curves are

the geodesics of some metric?

This problem gives rise to a certain closed system of partial

differential equations and hence to obstructions to finding such a

metric. It has been an open problem for at least 80 years. Surprisingly

it is harder in two dimensions than in higher dimensions. I shall present

a solution obtained jointly with Robert Bryant and Mike Eastwood.

In this talk I shall review analytical and numerical results on equilibrium configurations of rotating fluid bodies within Einstein's theory of gravitation.