Anti-self-dual conformal structures in neutral signature

Dunajski, Maciej; West, S. C.

doi:10.4171/051-1/4

Cited by 36 publications

(27 citation statements)

References 45 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Similar equations were written down on 4−dimensional Lorentzian manifolds [3]. Pseudo-Riemannian 4−manifolds with neutral signature are being studied by various authors from different point of view (see [2,4,8,10,11]). …”

Section: Introductionmentioning

confidence: 98%

Seiberg-Witten Equations on Pseudo-Riemannian Spinc Manifolds With Neutral Signature

Değirmenci

Karapazar

2012

Analele Universitatii "Ovidius" Constanta - Seria Matematica

View full text Add to dashboard Cite

Pseudo-Riemannian spin c manifolds were introduced by Ikemakhen in [7]. In the present work we consider pseudo-Riemannian 4−manifolds with neutral signature whose structure groups are SO+(2, 2). We prove that such manifolds have pseudo-Riemannian spin c structure. We construct spinor bundle S and half-spinor bundles S + and S − on these manifolds. For the first Seiberg-Witten equation we define Dirac operator on these bundles. Due to the neutral metric self-duality of a 2−form is meaningful and it enables us to write down second Seiberg-Witten equation. Lastly we write down the explicit forms of these equations on 4−dimensional flat space.

show abstract

Section: Introductionmentioning

confidence: 98%

Seiberg-Witten Equations on Pseudo-Riemannian Spinc Manifolds With Neutral Signature

Değirmenci

Karapazar

2012

Analele Universitatii "Ovidius" Constanta - Seria Matematica

View full text Add to dashboard Cite

show abstract

“…An almost para-hyperhermitian structure on a manifold consists of an almost parahypercomplex structure and a compatible semi-Riemannian metric necessarily of neutral signature. If all three structures involved in the definition of an almost para-hyperhermitian structure are parallel with respect to the Levi-Civita connection of the compatible metric, one arrives at the concept of para-hyper-Kähler structure, which is also referred to in the literature as neutral hyper-Kähler or hypersymplectic structure [4,7,10,13].…”

Section: Introductionmentioning

confidence: 99%

Para-hyperhermitian structures on tangent bundles

Vîlcu¹

2011

Proc. Estonian Acad. Sci.

View full text Add to dashboard Cite

In this paper we construct a family of almost para-hyperhermitian structures on the tangent bundle of an almost parahermitian manifold and study its integrability. Also, the necessary and sufficient conditions are provided for these structures to become para-hyper-Kähler.Key words: para-hyperhermitian structure, tangent bundle, paracomplex space form. PRELIMINARIESAn almost product structure on a smooth manifold M is a tensor field P of type (1,1) on M, P = ±Id, such thatwhere Id is the identity tensor field of type (1,1) on M.An almost para-hermitian structure on a differentiable manifold M is a pair (P, g), where P is an almost product structure on M and g is a semi-Riemannian metric on M satisfyingIn this case, (M, P, g) is said to be an almost para-hermitian manifold. It is easy to see that the dimension of M is even. Moreover, if ∇P = 0, then (M, P, g) is said to be a para-Kähler manifold.An almost complex structure on a smooth manifold M is a tensor field J of type (1,1) on M such thatAn almost para-hypercomplex structure on a smooth manifold M is a triple H = (J α ) α=1,3 , where J 1 is an almost complex structure on M and J 2 , J 3 are almost product structures on M, satisfyingIn this case (M, H) is said to be an almost para-hypercomplex manifold.A semi-Riemannian metric g on (M, H) is said to be compatible or adapted to the almost parahypercomplex structure H = (J α ) α=1,3 if it satisfies Remark 2.3. If (M, P, g) is an almost para-hermitian manifold, then we can define three tensor fields J 1 , J 2 , J 3 on T M by the equalities:It is easy to see that J 1 is an almost complex structure and J 2 , J 3 are almost product structures. We also have the following result (see [15]).

show abstract

“…A comprehensive survey of results on four-dimensional anti-self-dual metrics with the ultra-hyperbolic (neutral) signature was given by M. Dunajski and S. West in [4].…”

Section: Introductionmentioning

confidence: 99%

Lift of Invariant to Non-Invariant Solutions of Complex Monge-Ampère Equations

Sheftel¹,

Malykh

2008

JNMP

View full text Add to dashboard Cite

We show how partner symmetries of the elliptic and hyperbolic complex Monge-Ampère equations (CMA and HCMA) provide a lift of non-invariant solutions of three-and twodimensional reduced equations, i.e., a lift of invariant solutions of the original CMA and HCMA equations, to non-invariant solutions of the latter four-dimensional equations. The lift is applied to non-invariant solutions of the two-dimensional Helmholtz equation to yield non-invariant solutions of CMA, and to non-invariant solutions of three-dimensional wave equation and three-dimensional hyperbolic Boyer-Finley equation to yield non-invariant solutions of HCMA. By using these solutions as metric potentials, it may be possible to construct four-dimensional Ricci-flat metrics of Euclidean and ultra-hyperbolic signatures that have non-zero curvature tensors and no Killing vectors.

show abstract

Anti-self-dual conformal structures in neutral signature

Cited by 36 publications

References 45 publications

Seiberg-Witten Equations on Pseudo-Riemannian Spinc Manifolds With Neutral Signature

Seiberg-Witten Equations on Pseudo-Riemannian Spinc Manifolds With Neutral Signature

Para-hyperhermitian structures on tangent bundles

Lift of Invariant to Non-Invariant Solutions of Complex Monge-Ampère Equations

Contact Info

Product

Resources

About