(Conformally) semisymmetric spaces and special semisymmetric Weyl tensors

S. Brian Edgar

Semisymmetric spaces are a natural generalisation of symmetric spaces. For semisymmetric spaces in four dimensions with Lorentz signature, the Weyl tensor is easily seen (via spinors) to have a particularly simple quadratic property, which we call a special semisymmetric Weyl tensor. Using dimensionally dependent tensor identities, all (conformally) semisymmetric spaces are confirmed to have special semisymmetric Weyl tensors for all signatures in four dimensions. Furthermore, all Ricci-semisymmetric spaces with special semisymmetric Weyl tensors are shown to be semisymmetric for all signatures in four dimensions.
Counterexamples demonstrate that these two properties have no direct generalisations in higher dimensions.