Metrisability of two-dimensional projective structures
We carry out the programme of R. Liouville [19] to construct an explicit local obstruction to the existence of a Levi-Civita connection within a given projective structure [Γ] 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 [Γ] 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. © 2009 J. differential geometry.
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Published In
DOI
EISSN
ISSN
Publication Date
Volume
Issue
Start / End Page
Related Subject Headings
- General Mathematics
- 4904 Pure mathematics
- 0101 Pure Mathematics