Second Variation and Generalized Jacobi Equations for Curvature Invariants

We consider the second variation and the appropriate Jacobi eqautions for the scalar curvature and the quadratic curvature invariants based on an independent pair (g, T) formed by a metric and torsionless linear connection (so-called 'Palatini formalism'). The purely metric case is obtained as a consequence. The results are worked out in fill detail in view of applications to non-linear Lagrangian theories of gravitation.