Path geometries and almost Grassmann structures

Crampin, M.; Saunders, D.J.

doi:10.2969/aspm/04810225

Cited by 5 publications

(12 citation statements)

References 15 publications

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“…Regarding these curves as solutions to a system of (n − 1) second order ODEs, one can give an alternative definition of the path geometry as an equivalence class of systems of second order ODEs, where two systems are regarded as equivalent if they can be mapped into each other by a change of dependent and independent variables. In this paper we shall study three-dimensional path geometries encoded into a system of two second order ODEs Y ′′ = F (X, Y, Z, Y ′ , Z ′ ), Z ′′ = G(X, Y, Z, Y ′ , Z ′ ), (1.1) where Y ′ = dY /dX etc, and (F, G) are arbitrary functions on an open set in R 5 which we assume to be of class C 5 . Two such systems are locally equivalent if they are related by a point transformation (X, Y, Z) → (X(X, Y, Z), Y (X, Y, Z), Z(X, Y, Z)).…”

Section: Introductionmentioning

confidence: 99%

Twistor Geometry of a Pair of Second Order ODEs

Casey

Dunajski

Paul

2013

Commun. Math. Phys.

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Abstract. We discuss the twistor correspondence between path geometries in three dimensions with vanishing Wilczynski invariants and anti-self-dual conformal structures of signature (2, 2). We show how to reconstruct a system of ODEs with vanishing invariants for a given conformal structure, highlighting the Ricci-flat case in particular. Using this framework, we give a new derivation of the Wilczynski invariants for a system of ODEs whose solution space is endowed with a conformal structure. We explain how to reconstruct the conformal structure directly from the integral curves, and present new examples of systems of ODEs with point symmetry algebra of dimension four and greater which give rise to anti-self-dual structures with conformal symmetry algebra of the same dimension. Some of these examples are (2, 2) analogues of plane wave space-times in General Relativity. Finally we discuss a variational principle for twistor curves arising from the Finsler structures with scalar flag curvature.

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Section: Introductionmentioning

confidence: 99%

Twistor Geometry of a Pair of Second Order ODEs

Casey

Dunajski

Paul

2013

Commun. Math. Phys.

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“…We may construct T M using a technique described in [10]. We let VM be the manifold of equivalence classes…”

Section: Almost Grassmann Structuresmentioning

confidence: 99%

“…The fibre coordinates (u i ) on the new bundle are defined in terms of the fibre coordinates (y i ) of T M by u i = x 0 y i ; the quotient manifold may be thought of as the tensor product of the ordinary tangent bundle with the bundle of scalar densities of weight 1/(n + 1). The construction of the almost Grassmann structure may also be found in [10]. For any spray…”

Section: Almost Grassmann Structuresmentioning

confidence: 99%

Hilbert forms for a Finsler metrizable projective class of sprays

Crampin

Mestdag

Saunders

2013

Differential Geometry and its Applications

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The projective Finsler metrizability problem deals with the question whether a projective-equivalence class of sprays is the geodesic class of a (locally or globally defined) Finsler function. In this paper we use Hilbert-type forms to state a number of different ways of specifying necessary and sufficient conditions for this to be the case, and we show that they are equivalent. We also address several related issues of interest including path spaces, Jacobi fields, totally-geodesic submanifolds of a spray space, and the equivalence of path geometries and projective-equivalence classes of sprays.Comment: 23 page

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“…We have discussed a generalization of the construction of the Fefferman metrics from the Grassmanian point of view in [5], so we will not pursue this aspect of the matter any further here. We have discussed a generalization of the construction of the Fefferman metrics from the Grassmanian point of view in [5], so we will not pursue this aspect of the matter any further here.…”

Section: The Flat Casementioning

confidence: 99%

“…Incidentally, there is another way of approaching the definition of Q, which is to observe that 2 R 4 is a 6-dimensional real vector space, which can be identified with V in such a way that N defines the decomposable elements of 2 R 4 , and then Q becomes the Grassmanian of 2-planes in R 4 . We have discussed a generalization of the construction of the Fefferman metrics from the Grassmanian point of view in [5], so we will not pursue this aspect of the matter any further here.…”

Section: The Flat Casementioning

confidence: 99%

Fefferman-type metrics and the projective geometry of sprays in two dimensions

Crampin

Saunders

2007

Math. Proc. Camb. Phil. Soc.

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A spray is a second-order differential equation field on the slit tangent bundle of a differentiable manifold, which is homogeneous of degree 1 in the fibre coordinates in an appropriate sense; two sprays which are projectively equivalent have the same base-integral curves up to reparametrization. We show how, when the base manifold is two-dimensional, to construct from any projective equivalence class of sprays a conformal class of metrics on a four-dimensional manifold, of signature (2, 2); the Weyl conformal curvature of these metrics is simply related to the projective curvature of the sprays, and the geodesics of the sprays determine null geodesics of the metrics. The metrics in question have previously been obtained by Nurowski and Sparling (Classical and Quantum Gravity20 (2003) 4995–5016), by a different method involving the exploitation of a close analogy between the Cartan geometry of second-order ordinary differential equations and of three-dimensional Cauchy–Riemann structures. From this perspective the derived metrics are seen to be analoguous to those defined by Fefferman in the CR theory, and are therefore said to be of Fefferman type. Our version of the construction reveals that the Fefferman-type metrics are derivable from the scalar triple product, both directly in the flat case (which we discuss in some detail) and by a simple extension in general. There is accordingly in our formulation a very simple expression for a representative metric of the class in suitable coordinates.

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Path geometries and almost Grassmann structures

Cited by 5 publications

References 15 publications

Twistor Geometry of a Pair of Second Order ODEs

Twistor Geometry of a Pair of Second Order ODEs

Hilbert forms for a Finsler metrizable projective class of sprays

Fefferman-type metrics and the projective geometry of sprays in two dimensions

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