2008/01/01 by Robert L. Bryant, Maciej Dunajski, Bryant, Robert L. +3
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AP #math.DG #math.MP #nlin.SI
paper · pdf · doi:10.48550/arxiv.0801.0300
Minor corrections. Final version published in the Journal of Differential Geometry
arxiv created 2010/02/14 · arxiv updated 2010/02/26
We carry out the programme of R. Liouville \citeLiouville 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.