## Journal of Differential Geometry

### Metrisability of two-dimensional projective structures

#### 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

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