Connections and curvature in the Riemannian geometry of configuration spaces

Abstract Torsion free connections and a notion of curvature are introduced on the infinite dimensional nonlinear configuration space Γ of a Riemannian manifold M under a Poisson measure. This allows to state identities of Weitzenböck type and energy identities for anticipating stochastic integral operators. The one-dimensional Poisson case itself gives rise to a non-trivial geometry, a de Rham-Hodge-Kodaira operator, and a notion of Ricci tensor under the Poisson measure. The methods used in this paper have been thus far applied to ddimensional Brownian path groups, and rely on the introduction of a particular tangent bundle and associated damped gradient.