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Weakly conformally symmetric manifolds

Volume 150 / 2017

Carlo Alberto Mantica, Young Jin Suh Colloquium Mathematicum 150 (2017), 21-38 MSC: Primary 83C20; Secondary 53C40. DOI: 10.4064/cm6879s-1-2017 Published online: 11 September 2017

Abstract

We study the properties of weakly conformally symmetric pseudo-Riemannian manifolds, with particular emphasis on the $4$-dimensional Lorentzian case. We provide a decomposition of the conformal curvature tensor in dimensions $n \geq 5$. Moreover, some identities involving two particular covectors are stated; for example it is proven that under certain conditions the Ricci tensor and other tensors are Weyl compatible: this notion was recently introduced and investigated by Mantica and Molinari. Topological properties involving the vanishing of the first Pontryagin form are then stated. Further we study weakly conformally symmetric $4$-dimensional Lorentzian manifolds (space-times); it is proven that one of the previously defined covectors is null and unique up to scaling; moreover it is shown that under certain conditions the same vector is an eigenvector of the Ricci tensor and its integral curves are geodesics. Finally, it is shown that such a space-time is of Petrov type N with respect to the same vector.

Authors

  • Carlo Alberto ManticaPhysics Department Universitá degli Studi di Milano Via Celoria 16
    20133, Milano, Italy
    and
    I.I.S. Lagrange
    Via L. Modignani 65
    20161 Milano, Italy
    e-mail
  • Young Jin SuhDepartment of Mathematics Kyungpook National University Taegu 41566, Korea
    e-mail

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