Journal of Differential Geometry

Metrisability of two-dimensional projective structures

Robert Bryant, Maciej Dunajski, and Michael Eastwood

Full-text: Open access

Abstract

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.

Article information

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

Dates
First available in Project Euclid: 27 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1264601033

Digital Object Identifier
doi:10.4310/jdg/1264601033

Mathematical Reviews number (MathSciNet)
MR2581355

Zentralblatt MATH identifier
1196.53014

Citation

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. https://projecteuclid.org/euclid.jdg/1264601033


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