Weak Levi-Civita connection for the damped metric on the Riemannian path space and vanishing of Ricci tensor in adapted differential geometry

We shall establish in the context of adapted differential geometry on the p ath space P-mo (M) a Weitzenbock formula which generalizes that in (A. B. C ruzeiro and P. Malliavin, J. Funct. Anal. 177 (2000), 219-253), without hyp othesis on the Ricci tenser. The renormalized Ricci tensor will be vanished . The connection introduced in (A. B. Cruzeiro and S. Fang, 1997, J. Funct. Anal, 143, 400 414) will play a central role. (C) 2001 Academic Press.