Metrisability of two-dimensional projective structures

Robert Bryant The Mathematical Sciences Research Institute Maciej Dunajski University of Cambridge Michael Eastwood Australian National University

Differential Geometry mathscidoc:1609.10147

Journal of Differential Geometry, 83, (3), 465-499, 2009
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 [..] 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 [..] 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.
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  author={Robert Bryant, Maciej Dunajski, and Michael Eastwood},
  booktitle={Journal of Differential Geometry},
Robert Bryant, Maciej Dunajski, and Michael Eastwood. METRISABILITY OF TWO-DIMENSIONAL PROJECTIVE STRUCTURES. 2009. Vol. 83. In Journal of Differential Geometry. pp.465-499.
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