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Publications [#243378] of Robert Bryant
search www.ams.org.Papers Published
 with Bryant, R; Dunajski, M; Eastwood, M, Metrisability of twodimensional projective structures,
Journal of Differential Geometry, vol. 83 no. 3
(2009),
pp. 465500, ISSN 0022040X [MR2581355], [arXiv:0801.0300v1 [math.DG]], [doi]
(last updated on 2018/10/17)
Abstract: We carry out the programme of R. Liouville \cite{Liouville} to construct an
explicit local obstruction to the existence of a LeviCivita 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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