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  • 1.
    Andersson, Fredrik
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Edgar, S.B.
    Local existence of symmetric spinor potentials for symmetric (3,1)-spinors in Einstein space-times2001In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 37, no 4, p. 273-290Article in journal (Refereed)
    Abstract [en]

    We investigate the possibility of existence of a symmetric potential HABA'B'=H(AB)(A'B') for a symmetric (3,1)-spinor LABCA', e.g., a Lanczos potential of the Weyl spinor, as defined by the equation LABCA'=?(AB'H BC)A'B'. We prove that in all Einstein space-times such a symmetric potential HABA'B' exists. Potentials of this type have been found earlier in investigations of some very special spinors in restricted classes of space-times. A tensor version of this result is also given. We apply similar ideas and results by Illge to Maxwell's equations in a curved space-time. © 2001 Elsevier Science B.V.

  • 2.
    Edgar, Brian
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Proofs of existence of local potentials for trace-free symmetric 2-forms using dimensionally dependent identities2005In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 54, no 3, p. 251-261Article in journal (Refereed)
    Abstract [en]

    We exploit four-dimensional tensor identities to give a very simple proof of the existence of a Lanczos potential for a Weyl tensor in four dimensions with any signature, and to show that the potential satisfies a simple linear second-order differential equation, e.g., a wave equation in Lorentz signature. Furthermore, we exploit higher-dimensional tensor identities to obtain the analogous results for (m, m)-forms in 2m dimensions. © 2004 Elsevier B.V. All rights reserved.

  • 3.
    Gomez-Lobo, A.G.-P.
    et al.
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Valiente, Kroon J.A.
    Valiente Kroon, J.A., School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London, E1 4NS, United Kingdom.
    Killing spinor initial data sets2008In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 58, no 9, p. 1186-1202Article in journal (Refereed)
    Abstract [en]

    A 3+1 decomposition of the twistor and valence-2 Killing spinor equation is made using the space-spinor formalism. Conditions on initial data sets for the Einstein vacuum equations are given so that their developments contain solutions to the twistor and/or Killing equations. These lead to the notions of twistor and Killing spinor initial data. These notions are used to obtain a characterisation of initial data sets whose developments are of Petrov type N or D. © 2008 Elsevier B.V. All rights reserved.

  • 4.
    Jonasson, Jens
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Multiplication of solutions for linear overdetermined systems of partial differential equations2008In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 58, no 8, p. 1015-1029Article in journal (Refereed)
    Abstract [en]

    A large family of linear, usually overdetermined, systems of partialdifferential equations that admit a multiplication of solutions, i.e, a bilinearand commutative mapping on the solution space, is studied. Thisfamily of PDE’s contains the Cauchy–Riemann equations and the cofactorpair systems, included as special cases. The multiplication provides amethod for generating, in a pure algebraic way, large classes of non-trivialsolutions that can be constructed by forming convergent power series oftrivial solutions.

  • 5.
    Marciniak, Krzysztof
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Science and Technology.
    Blaszak, Maciej
    Insitution för fysik Adam Mickiewicz universitet, Poznan, Poland.
    Non-Hamiltonian systems separable by Hamilton-Jacobi method2008In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 58, no 5, p. 557-575Article in journal (Refereed)
    Abstract [en]

    We show that with every separable classical Stäckel system of Benenti type on a Riemannian space one can associate, by a proper deformation of the metric tensor, a multi-parameter family of non-Hamiltonian systems on the same space, sharing the same trajectories and related to the seed system by appropriate reciprocal transformations. These systems are known as bi-cofactor systems and are integrable in quadratures as the seed Hamiltonian system is. We show that with each class of bi-cofactor systems a pair of separation curves can be related. We also investigate the conditions under which a given flat bi-cofactor system can be deformed to a family of geodesically equivalent flat bi-cofactor systems. © 2007 Elsevier Ltd. All rights reserved.

  • 6.
    Senovilla, Jose M M
    et al.
    Physics University of the Basque Country.
    Edgar, Brian
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    A weighted de Rham operator acting on arbitrary tensor fields and their local potentials2006In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 56, no 10, p. 2135-2162Article in journal (Refereed)
    Abstract [en]

    We introduce a weighted de Rham operator which acts on arbitrary tensor fields by considering their structure as r-fold forms. We can thereby define associated superpotentials for all tensor fields in all dimensions and, from any of these superpotentials, we deduce in a straightforward and natural manner the existence of 2 r potentials for any tensor field, where r is its form-structure number. By specialising this result to symmetric double forms, we are able to obtain a pair of potentials for the Riemann tensor, and a single (2, 3)-form potential for the Weyl tensor due to its tracelessness. This latter potential is the n-dimensional version of the double dual of the classical four-dimensional (2, 1)-form Lanczos potential. We also introduce a new concept of harmonic tensor fields, and demonstrate that the new weighted de Rham operator has many other desirable properties and, in particular, is the natural operator to use in the Laplace-like equation for the Riemann tensor. © 2005 Elsevier Ltd. All rights reserved.

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