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It is well-known that Einstein gravity is kinematic (no non-trivial vacuumsolution;i.e. Riemann vanishes whenever Ricci does so) in $3$ dimension becauseRiemann is entirely given in terms of Ricci. Could this property beuniversalized for all odd dimensions in a generalized theory? The answer isyes, 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 Riemanntensor 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 andit uniquely picks out pure Lovelock gravity from all other generalizations ofEinstein gravity. It serves as a good discerning and guiding criterion forgravitational equation in higher dimensions.

A discerning gravitational property for gravitational equation in higher dimensions