Journal of Differential Geometry

Metrisability of two-dimensional projective structures

Robert Bryant, Maciej Dunajski, and Michael Eastwood

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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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J. Differential Geom., Volume 83, Number 3 (2009), 465-500.

First available in Project Euclid: 27 January 2010

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Bryant, Robert; Dunajski, Maciej; Eastwood, Michael. Metrisability of two-dimensional projective structures. J. Differential Geom. 83 (2009), no. 3, 465--500. doi:10.4310/jdg/1264601033.

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