Publications [#141021] of Robert L Bryant

Preprints

1. with M. Dunajski, M. Eastwood, Metrisability of two-dimensional projective structures, J. Differential Geometry, vol. 83 no. 3 (2009), pp. 465--499, ISSN 0022-040X
   (last updated on 2010/11/19)

   Abstract:
   We carry out the programme of R. Liouville to construct an explicit local obstruction to the existence of a Levi--Civita connection within a given projective structure $[\Gamma]$ on a surface. The obstruction is of order 5 in the components of a connection in a projective class. It can be expressed as a point invariant for a second order ODE whose integral curves are the geodesics of $[\Gamma]$ or as a weighted scalar projective invariant of the projective class. If the obstruction vanishes we find the sufficient conditions for the existence of a metric in the real analytic case. In the generic case they are expressed by the vanishing of two invariants of order 6 in the connection. In degenerate cases the sufficient obstruction is of order at most 8.