G_{2} CARTAN CONNECTION ASSOCIATED WITH EQUATION z′ = F (x, y, y′, y′′, z)

PAWEL NUROWSKI (WARSAW UNIVERSITY)

Abstract. In 1912 D. Hilbert observed that the general solution of the equation z′ = (y′′)^{2} can not be expressed without integrals. Later E. Cartan solved the equivalence problem for the geometry associated with equation

(*)  z′ = F (x, y, y′, y′′, z).

He showed that only equations satisfying F_{(}y′′y′′) = 0 have general solution without integrals. In the F_{y′′y′′} ≠ 0 case he constructed all the local invariants for (*) and showed that they are expressible in terms of Cartan connection with values in the Lie algebra of the noncompact form of the exceptional group G_{2}. The curvature of this connection turns out to be flat only in the case when (*) is equivalent to the Hilbert example.

In the lecture we briefly describe the Cartan approach to the problem and show an alternative description of the invariants in terms of a conformal geometry of signature (+, +, −, −, −) which is naturally associated with the equation (*) satisfying F_{y′′y′′} ≠ 0.