Connections and curvature in the Riemannian geometry
of configuration spaces
| Nicolas Privault |
| Université de la Rochelle |
| Avenue Marillac |
| 17042 La Rochelle Cedex 1 |
| France |
Abstract:
Torsion free connections
and a notion of curvature are
introduced on
the configuration space GammaM of a Riemannian manifold M
under a Poisson measure, which is
an infinite-dimensional nonlinear space.
This allows to state
identities of Weitzenböck-Bochner type
and energy identities for anticipating stochastic
integral operators.
The one-dimensional Poisson case itself
gives rise to a non-trivial Riemannian geometry,
a de Rham-Hodge-Kodaira operator
and a notion of Ricci tensor under the
Poisson measure,
which are obtained from methods
of d-dimensional Brownian path groups.
This construction relies on the introduction
of a particular tangent bundle
and its associated damped gradient.
We point out differences and similarities between the geometries
of configuration and path spaces.
Key words: Configuration spaces, covariant derivatives,
connections.
Mathematics Subject Classification (1991):
60H07, 58G32, 53B21.
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