The Divergence on Submanifolds of the Wiener Space

Using Malliavin′s calculus, the divergence, the covariant derivative, and the Riemann and Ricci curvatures of a submanifold of the Wiener space are defined. It is shown that the Ricci and Riemann curvatures appear in the commutator of the divergence operator and covariant derivative operator. Capacities are used to restrict the commutator formula to a submanifold and to deduce that the curvature belongs to all Lp(daξ)-spaces where aξ is the area measure associated to the submanifold. A version of Weitzenböck′s formula, a Bochner type formula, and a criterion for the existence of the divergence are proved. A generalization to higher dimensional tensor fields is stated, and a connection with stochastic integration on the submanifold of the Wiener space is worked out.