Some aspects of field equations in generalized theories of gravity

A class of theories of gravity based on a Lagrangian $L=L(R^a{}_{bcd},g^{ab})$ which depends on the curvature and metric—but not on the derivatives of the curvature tensor—is of interest in several contexts including in the development of the paradigm that treats gravity as an emergent phenomenon. This class of models contains, as an important subset, all Lanczos-Lovelock models of gravity. I derive several identities and properties which are useful in the study of these models and clarify some of the issues that seem to have received insufficient attention in the past literature.