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We classify all the six derivative Lagrangians of gravity, whose traced fieldequations are of second or third order, in arbitrary dimensions. In the formercase, the Lagrangian in dimensions greater than six, reduces to an arbitrarylinear combination of the six dimensional Euler density and the two linearlyindependent cubic Weyl invariants. In five dimensions, besides the independentcubic Weyl invariant, we obtain an interesting cubic combination, whose fieldequations for static spherically symmetric spacetimes are of second order. Inthe later case, in arbitrary dimensions we obtain two combinations, which indimension three, are equivalent to the complete contraction of two Cottontensors. Moreover, we also recover all the conformal anomalies in sixdimensions. Finally, we present some static, spherically symmetric solutionsfor these Lagrangians.

Classification of Six Derivative Lagrangians of Gravity and Static Spherically Symmetric Solutions