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Keywords
(7)
Degeneration
Differential Geometry
Integrable System
Mathematical Analysis
Projective Structure
Vanishing of
Second Order
Related Publications
(3)
Geodesic equivalence and integrability
Trajectory equivalence and corresponding integrals
Statefinder̵...
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Metrisability of twodimensional projective structures
Metrisability of twodimensional projective structures,Robert L. Bryant,Maciej Dunajski,Michael Eastwood
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Metrisability of twodimensional projective structures
(
Citations: 21
)
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Robert L. Bryant
,
Maciej Dunajski
,
Michael Eastwood
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.
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.
Fourdimensional 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 pseudoRiemannian 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 pseudoRiemannian 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 antiselfdual manifolds, which vanishes when the conformal class contains a neutral K¨ahler metric with conformal null symmetry...
Johann Davidov
,
et al.
Geometry of neutral metrics in dimesnion four
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Citations
(21)
Geodesically equivalent metrics in general relativity
(
Citations: 1
)
Vladimir S. Matveev
Journal:
Journal of Geometry and Physics  J GEOM PHYSICS
, 2011
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