Quadratic Curvature Gravity with Second Order Trace and Massive Gravity Models in Three Dimensions

The quadratic curvature lagrangians having metric field equations with second order trace are constructed relative to an orthonormal coframe. In $n>4$ dimensions, pure quadratic curvature lagrangian having second order trace constructed contains three free parameters in the most general case. The fourth order field equations of some of these models, in arbitrary dimensions, are cast in a particular form using the Schouten tensor. As a consequence, the field equations for the New massive gravity theory are related to those of the Topologically massive gravity. In particular, the conditions under which the latter is "square root" of the former are presented.