The n-dimensional Lorentzian manifolds with vanishing second covariant derivative of the Riemann tensor—2–symmetric spacetimes—are characterized and classified. The main result is that either they are locally symmetric or they have a covariantly constant null vector field, in this case defining a subfamily of Brinkmann's class in n dimensions. Related issues and applications are considered, and new open questions presented.

http://iopscience.iop.org/article/10.1088/0264-9381/25/24/245011/pdf

On generalized curvature tensors on some Riemannian manifolds

Second-order symmetric Lorentzian manifolds: I. Characterization and general results

We investigate curvature properties of hypersurfaces of a semi-Rieman- nian space form satisfyingR·C = LQ(S,C), which is a curvature condition of pseu- dosymmetry type. We prove that under some additional assumptions the ambient space of such hypersurfaces must be semi-Euclidean and that they a re quasi-Einstein Ricci- semisymmetricmanifolds.

https://www.impan.pl/shop/publication/transaction/download/product/87805?download.pdf

Quasi-Einstein hypersurfaces in semi-Riemannian space forms

/pdf/on-the-equivalence-of-the-ricci-pseudosymmetry-and-1zm7htv18r.pdf

On the equivalence of the Ricci-pseudosymmetry and pseudosymmetry

By a continuum we usually mean a metric (or Hausdorff) compact connected space. The original definition of 1883, due to Georg Cantor, [126], p. 576, stated that a subset of a Euclidean space is a continuum provided it is perfect (i.e. closed and dense-in-itself, or — equivalently — coincides with its first derivative) and connected, i.e. if for every two of its points a and b and for each positive number є there corresponds a finite system of points a = p 0, p 1, ..., p n = b such that the distance between any two consecutive points of the system is less than є. The equivalence of the two definitions for compact metric spaces is shown e.g. in Kuratowski’s monograph [390], vol. 2, §47, I, Theorem 0, p. 167.

History of Continuum Theory

On semisymmetric para-Kähler manifolds

/pdf/on-generalized-curvature-tensors-on-some-riemannian-2wgj293via.pdf

On generalized curvature tensors on some Riemannian manifolds

Citations

Second-order symmetric Lorentzian manifolds: I. Characterization and general results

Quasi-Einstein hypersurfaces in semi-Riemannian space forms

On the equivalence of the Ricci-pseudosymmetry and pseudosymmetry

History of Continuum Theory

On semisymmetric para-Kähler manifolds

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