I want to answer your first question. The concept of symmetric curvature tensor is defined by N.Boroojerdian and A. Heidari. This concept is related to the notions divergence, Laplacian and Ricci Tensor. This curvature tensor is responsible for many behaviours of a geomteric space. see: "SYMMETRIC CURVATURE TENSOR,Bulletin of the Iranian Mathematical Society Vol. 37 No. 3 (2011), pp 249-267".