A second-order identity for the Riemann tensor and applications
  A second-order differential identity for the Riemann tensor is obtained, on a manifold with symmetric connection. Several old and some new differential identities for the Riemann and Ricci tensors descend from it. Applications to manifolds with Recurrent or Symmetric structures are discussed. The new structure of K-recurrency naturally emerges from an invariance property of an old identity by Lovelock. 
Carlo Alberto Mantica and Luca Guido Molinari