Metrisability of two-dimensional projective structures
Abstract
We carry out the programme of R. Liouville \cite{Liouville} 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.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2008
- DOI:
- 10.48550/arXiv.0801.0300
- arXiv:
- arXiv:0801.0300
- Bibcode:
- 2008arXiv0801.0300B
- Keywords:
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- Mathematics - Differential Geometry;
- Mathematical Physics;
- Mathematics - Analysis of PDEs;
- Nonlinear Sciences - Exactly Solvable and Integrable Systems
- E-Print:
- Minor corrections. Final version published in the Journal of Differential Geometry