Publications [#243378] of Robert Bryant

Papers Published

1. with Bryant, R; Dunajski, M; Eastwood, M, Metrisability of two-dimensional projective structures, Journal of Differential Geometry, vol. 83 no. 3 (January, 2009), pp. 465-500, International Press of Boston, ISSN 0022-040X [MR2581355], [arXiv:0801.0300v1 [math.DG]], [doi]
   (last updated on 2025/02/21)

   Abstract:
   We carry out the programme of R. Liouville [19] to construct an explicit local obstruction to the existence of a Levi-Civita connection within a given projective structure [Γ] 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 [Γ] 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. © 2009 J. differential geometry.