  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.
