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Connections and Curvature in the Riemannian Geometry of Configuration Spaces

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 us to state identities of Weitzenbock 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 d-dimensional Brownian path groups and rely on the introduction of a particular tangent bundle and associated damped gradient.