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Metrisability of two-dimensional projective structures

# Metrisability of two-dimensional projective structures,Robert L. Bryant,Maciej Dunajski,Michael Eastwood

Metrisability of two-dimensional projective structures
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.
Published in 2008.
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## Citation Context (11)

• ...In [3] necessary and sufficient conditions have been determined in the case when dim U = 2 for the existence of a (pseudo) Riemannian metric on U whose geodesics coincide with the geodesics of the given projective structure...
• ...If n = 2 this conformal structure is necessarily ASD [12, 6] and we shall show (Theorem 4.1) that the projective structure on a surface U is metrisable if and only if the induced conformal structure on T U admits a K¨ahler metric or a para–K¨ahler metric. This establishes a conjecture made in [3]...
• ... Thus it fits into the classification [6] as explained in [3]...
• ...The components of YC ′ are given by (2.14) in [3] and referred to as Liouville’s invariants...
• ...In [3] it was shown that a projective structure comes from a (possibly Lorentzian) metric on U if and only if there exists a covariantly constant section (σ A...
• ...= 0. (4.48) (This is (7.46) or (3.20) in [3])...
• ...But we know that these will hold automatically for (4.43): in [3] it was shown that the first constraint arises after taking two derivatives of (4.48)...

### Maciej Dunajski, et al. Four-dimensional metrics conformal to Kähler

• ...One can nd more historical details in the surveys [3, 9, 37] and in the introductions to the papers [25, 26, 27, 28, 32, 34, 35, 50]...

### Alexey V. Bolsinov, et al. A Fubini theorem for pseudo-Riemannian geodesically equivalent metrics

• ...We aim to clarify this procedure (which appears to have been initiated in [7] and developed by [8] and is closely related to the Statefinder approach of [12]) and link it to some recent work [3] on the metrisability of projective structures...
• ...recently [3]. They come down to vanishing of three weighted invariants of differential order at most 8 in connection defining the projective structure...
• ...Examining the obstructions of [3] shows that the projective structure defined by (4.13) is metrisable for any f(a) (and therefore for any choice of the Lagrangian L). Following the algorithm of [3] leads to the expression for the metric...
• ...Examining the obstructions of [3] shows that the projective structure defined by (4.13) is metrisable for any f(a) (and therefore for any choice of the Lagrangian L). Following the algorithm of [3] leads to the expression for the metric...
• ...In [3] the consistency conditions for this system where found in terms of point invariants of the associated second order ODE (A1)...
• ...The details of this construction and the invariants themselves are rather complicated and we refer the reader to [3]...

### Maciej Dunajski, et al. Cosmic jerk, snap and beyond

• ...One can find more historical details in the surveys [3, 40, 9] and in the introduction to the papers [37, 30, 29, 53, 34, 36, 28, 27]...

### Alexey V. Bolsinov, et al. Fubini Theorem for pseudo-Riemannian metrics

• ...We should also mention the recent work by Bryant, Dunajski and Eastwood [12] on two dimensional metrisable projective structures which suggests the existence of a conformal invariant on neutral anti-self-dual manifolds, which vanishes when the conformal class contains a neutral K¨ahler metric with conformal null symmetry...

## References (10)

### Prolongations of Geometric Overdetermined Systems(Citations: 25)

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### Exterior differential systems(Citations: 251)

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### Invariant characterization of Liouville metrics and polynomial integrals(Citations: 16)

Journal: Journal of Geometry and Physics - J GEOM PHYSICS , vol. 58, no. 8, pp. 979-995, 2008
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