Markovian Connection, Curvature and Weitzenbِck Formula on Riemannian Path Spaces

We shall consider on a Riemannian path space Pmo(M) the Cruzeiro–Malliavinʹs Markovian connection. The Laplace operator will be defined as the divergence of the gradient. We shall compute explicitly the associated curvature tensor. A Weitzenböck formula will be established. To this end, we shall introduce an “inner product” between the tangent processes and simple vector fields.