Intrinsic geometry of oriented congruences in three dimensions

Hill, C. Denson; Nurowski, Paweł

doi:10.1016/j.geomphys.2008.10.001

Cited by 6 publications

(4 citation statements)

References 14 publications

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“…In [10], he and his collaborators use the CR function in such a structure to create a very appropriate choice of coordinates for a twisting type N Einstein space, and reduce the Einstein equations to a set of nonlinear PDEs for a couple of functions of three variables. Then in [11], he makes a clever ansatz depending only on a single variable and discovers a particular twisting solution; unfortunately, that solution turns out to be the same as the one mentioned above first found by Leroy, as he notes in a more recent paper [13]. However, we were quite intrigued by the approach and have made some efforts to follow it through with the hope of obtaining more general solutions of the equations in Nurowski's article.…”

Section: Introductionmentioning

confidence: 83%

Lower order ODEs to determine new twisting type N Einstein spaces via CR geometry

Zhang

Finley

2012

Class. Quantum Grav.

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In the search for vacuum solutions, with or without a cosmological constant, Λ, of the Einstein field equations of Petrov type N with twisting principal null directions, the CR structures to describe the parameter space for a congruence of such null vectors provide a very useful tool. Work of Hill, Lewandowski and Nurowski has given a good foundation for this, reducing the field equations to a set of differential equations for two functions, one real and one complex, of three variables. Under the assumption of the existence of one Killing vector, the (infinite-dimensional) classical symmetries of those equations are determined and group-invariant solutions are considered. This results in a single ODE of the third order which may easily be reduced to one of the second order. A one-parameter class of power series solutions, g(w), of this second-order equation is realized, holomorphic in a neighborhood of the origin and behaving asymptotically as a simple quadratic function plus lower order terms for large values of w, which constitutes new solutions of the twisting type N problem. The solution found by Leroy, and also by Nurowski, is shown to be a special case in this class. Cartan’s method for determining equivalence of CR manifolds is used to show that this class is indeed much more general. In addition, for a special choice of a parameter, this ODE may be integrated once to provide a first-order Abel equation. It can also determine new solutions to the field equations, although no general solution has yet been found for it.

show abstract

Section: Introductionmentioning

confidence: 83%

Lower order ODEs to determine new twisting type N Einstein spaces via CR geometry

Zhang

Finley

2012

Class. Quantum Grav.

View full text Add to dashboard Cite

show abstract

“…In [10], he and his collaborators use the CR function in such a structure to create a very appropriate choice of coordinates for a twisting type N Einstein space, and reduce the Einstein equations to a set of nonlinear PDE's for a couple of functions of three variables. Then in [11], he makes a clever ansatz depending only on a single variable and discovers a particular twisting solution; unfortunately that solution turns out to be the same as the one mentioned above as first found by Leroy, as he notes in a more recent paper [13]. However, we were quite intrigued by the approach and have made some efforts to follow it through with the hopes of obtaining more general solutions of the equations in Nurowski's article.…”

Section: Introductionmentioning

confidence: 83%

Lower-order ODEs to determine new twisting type N Einstein spaces via CR geometry

Zhang,

Finley

2011

Preprint

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In the search for vacuum solutions, with or without a cosmological constant, Λ, of the Einstein field equations of Petrov type N with twisting principal null directions, the CR structures to describe the parameter space for a congruence of such null vectors provide a very useful tool. A work of Hill, Lewandowski and Nurowski has given a good foundation for this, reducing the field equations to a set of differential equations for two functions, one real, one complex, of three variables. Under the assumption of the existence of one Killing vector, the (infinite-dimensional) classical symmetries of those equations are determined and group-invariant solutions are considered. This results in a single ODE of the third order which may easily be reduced to one of the second order. A one-parameter class of power series solutions, g(w), of this second-order equation is realized, holomorphic in a neighborhood of the origin and behaving asymptotically as a simple quadratic function plus lower-order terms for large values of w, which constitutes new solutions of the twisting type N problem. The solution found by Leroy, and also by Nurowski, is shown to be a special case in this class. Cartan's method for determining equivalence of CR manifolds is used to show that this class is indeed much more general.In addition, for a special choice of a parameter, this ODE may be integrated once, to provide a first-order Abel equation. It can also determine new solutions to the field equations although no general solution has yet been found for it.

show abstract

“…Besides the half conformally flat metrics and the conformally Einstein ones, there are few known examples of strictly Bach-flat manifolds, meaning the ones which are neither half conformally flat nor conformally Einstein (see, for example, [1,23,26]). Motivated by this lack of examples, we first construct new explicit four-dimensional Bach-flat manifolds of neutral signature.…”

Section: Preliminariesmentioning

confidence: 99%