  In the Riemann geometry, the metric's equation of motion for an arbitrary
Lagrangian is succinctly expressed in term of the first variation of the action
with respect to the Riemann tensor if the Riemann tensor were independent of
the metric. Let this variation be called the E-tensor. Noting that the E-tensor
and equations of the motion for a general Lovelock gravity have the same
differential degree, we define generalized Lovelock gravity as polynomial
scalar densities constructed out from the Riemann tensor and its arbitrary
covariant derivatives such that they lead to the same differential degree for
the E-tensor and the metric's equation of motion. We consider Lagrangian
densities which are functional of the metric and the first covariant derivative
of the Riemann tensor. We then present the first non-trivial examples of the
generalized Lovelock gravity terms.
