On Lovelock analogs of the Riemann tensor

Abstract

It is possible to define an analog of the Riemann tensor for Nth order Lovelock gravity, its characterizing property being that the trace of its Bianchi derivative yields the corresponding analog of the Einstein tensor. Interestingly there exist two parallel but distinct such analogs and the main purpose of this note is to reconcile both formulations. In addition we will introduce a simple tensor identity and use it to show that any pure Lovelock vacuum in odd $$d=2N+1$$ dimensions is Lovelock flat, i.e. any vacuum solution of the theory has vanishing Lovelock–Riemann tensor. Further, in the presence of cosmological constant it is the Lovelock–Weyl tensor that vanishes.