Open Access
November 2009 Metrisability of two-dimensional projective structures
Robert Bryant, Maciej Dunajski, Michael Eastwood
J. Differential Geom. 83(3): 465-500 (November 2009). DOI: 10.4310/jdg/1264601033


We carry out the programme of R. Liouville, Sur les invariants de certaines équations différentielles et sur leurs applications, 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.


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Robert Bryant. Maciej Dunajski. Michael Eastwood. "Metrisability of two-dimensional projective structures." J. Differential Geom. 83 (3) 465 - 500, November 2009.


Published: November 2009
First available in Project Euclid: 27 January 2010

zbMATH: 1196.53014
MathSciNet: MR2581355
Digital Object Identifier: 10.4310/jdg/1264601033

Rights: Copyright © 2009 Lehigh University

Vol.83 • No. 3 • November 2009
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