On Lovelock analogues of the Riemann tensor

TL;DR

This paper explores two formulations of the Lovelock Riemann tensor, showing their equivalence, and demonstrates that vacuum solutions in odd dimensions are Lovelock flat, with the Lovelock-Weyl tensor vanishing when a cosmological constant is present.

Contribution

It reconciles two parallel formulations of the Lovelock Riemann tensor and analyzes vacuum solutions in Lovelock gravity in odd dimensions.

Findings

Two distinct Lovelock Riemann tensor analogues are shown to be reconcilable.

Pure Lovelock vacuum solutions in odd dimensions are Lovelock flat.

Lovelock-Weyl tensor vanishes in the presence of a cosmological constant.

Abstract

It is possible to define an analogue of the Riemann tensor for $N$th order Lovelock gravity, its characterizing property being that the trace of its Bianchi derivative yields the corresponding analogue of the Einstein tensor. Interestingly there exist two parallel but distinct such analogues and the main purpose of this note is to reconcile both these formulations. In addition we will 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 presence of cosmological constant it is the Lovelock-Weyl tensor that vanishes.