Weakly conformally symmetric manifolds

Carlo Alberto Mantica¹ ; Young Jin Suh²

¹ Physics Department Universitá degli Studi di Milano Via Celoria 16 20133, Milano, Italy and I.I.S. Lagrange Via L. Modignani 65 20161 Milano, Italy
² Department of Mathematics Kyungpook National University Taegu 41566, Korea

Colloquium Mathematicum, Tome 150 (2017) no. 1, pp. 21-38.

Résumé

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.

Détail

DOI : 10.4064/cm6879s-1-2017

Keywords: study properties weakly conformally symmetric pseudo riemannian manifolds particular emphasis dimensional lorentzian provide decomposition conformal curvature tensor dimensions geq moreover identities involving particular covectors stated example proven under certain conditions ricci tensor other tensors weyl compatible notion recently introduced investigated mantica molinari topological properties involving vanishing first pontryagin form stated further study weakly conformally symmetric dimensional lorentzian manifolds space times proven previously defined covectors null unique scaling moreover shown under certain conditions vector eigenvector ricci tensor its integral curves geodesics finally shown space time petrov type respect vector