  It is well-known that Einstein gravity is kinematic (no non-trivial vacuum
solution;i.e. Riemann vanishes whenever Ricci does so) in $3$ dimension because
Riemann is entirely given in terms of Ricci. Could this property be
universalized for all odd dimensions in a generalized theory? The answer is
yes, and this property uniquely singles out pure Lovelock (it has only one
$N$th order term in action) gravity for which $N$th order Lovelock Riemann
tensor is indeed given in terms of corresponding Ricci for all odd $d=2N+1$
dimensions. This feature of gravity is realized only in higher dimensions and
it uniquely picks out pure Lovelock gravity from all other generalizations of
Einstein gravity. It serves as a good discerning and guiding criterion for
gravitational equation in higher dimensions.
