In order to study the properties of Lovelock gravity theories in low
dimensions, we define the kth-order Riemann-Lovelock tensor as a certain
quantity having a total 4k-indices, which is kth-order in the Riemann curvature
tensor and shares its basic algebraic and differential properties. We show that
the kth-order Riemann-Lovelock tensor is determined by its traces in dimensions
2k \le D <4k. In D=2k+1 this identity implies that all solutions of pure
kth-order Lovelock gravity are `Riemann-Lovelock' flat. It is verified that the
static, spherically symmetric solutions of these theories, which are missing
solid angle space times, indeed satisfy this flatness property. This
generalizes results from Einstein gravity in D=3, which corresponds to the k=1
case. We speculate about some possible further consequences of Riemann-Lovelock
curvature.Comment: 12 page
