GL(2,R) geometry of ODE's

Abstract

We study five dimensional geometries associated with the 5-dimensional irreducible representation of GL(2,R). These are special Weyl geometries in signature (3,2) having the structure group reduced from CO(3,2) to GL(2,R). The reduction is obtained by means of a conformal class of totally symmetric 3-tensors. Among all GL(2,R) geometries we distinguish a subclass which we term `nearly integrable GL(2,R)geometries'. These define a unique gl(2,R) connection which has totally skew symmetric torsion. This torsion splits onto the GL(2,R) irreducible components having respective dimensions 3 and 7. We prove that on the solution space of every 5th order ODE satisfying certain three nonlinear differential conditions there exists a nearly integrable GL(2,R) geometry such that the skew symmetric torsion of its unique gl(2,R) connection is very special. In contrast to an arbitrary nearly integrable GL(2,R) geometry, it belongs to the 3-dimensional irreducible representation of GL(2,R). The conditions for the existence of the structure are lower order equivalents of the Doubrov-Wilczynski conditions found recently by Boris Doubrov [7]. We provide nontrivial examples of 5th order ODEs satisfying the three nonlinear differential conditions, which in turn provides examples of inhomogeneous GL(2,R) geometries in dimension five, with torsion in R^3. We also outline the theory and the basic properties of GL(2,R) geometries associated with n-dimensional irreducible representations of GL(2,R) in 5<n<10. In particular we give conditions for an n-th order ODE to possess this geometry on its solution space.