On the geometry of profinite diffeological spaces
Anahita Eslami-Rad, Jean-Pierre Magnot, Enrique G. Reyes
TL;DR
This work develops a comprehensive framework for profinite diffeological spaces, i.e. projective limits of finite-dimensional manifolds endowed with a projective diffeology, to study infinite-dimensional geometric objects in a smooth setting. By introducing notions such as the maximal profinite diffeology, cylindrical and tame differential forms, and compatible metric and symplectic structures, the paper shows how tangent/cotangent data and differential complexes descend to the limit while avoiding common pathologies. It proves equivalences among tangent notions under finite cylindricality, constructs $\\Omega_t^n(E_A)$ with a well-defined exterior differential, and analyzes non-degeneracy and Hamiltonian geometry in this profinite context. A suite of explicit examples (matrices, jet spaces, Wiener space) demonstrates the practicality of the framework for both analysis and geometry, including stochastic-geometric settings via cylindrical evaluations and Malliavin-type calculus. Overall, the results unify and extend differential-geometric constructions to broad infinite-dimensional limits, providing tools applicable to analysis on function spaces, stochastic geometry, and related areas.
Abstract
We consider the class of profinite diffeological spaces, that is, diffeological spaces which diffeologies are deduced by pull-back of diffeologies on finite-dimensional manifolds through a system of projection mappings. This class includes inductive limits of finite-dimensional manifolds, as well as solution spaces of differential relations and spaces that agree with cylindrical approximations. We analyze tangent and cotangent spaces, differential forms, Riemannian metrics, connections, differential operators, Laplacians, symplectic forms, momentum maps and the relation between de Rham and singular cohomology on them, unifying constructions partially developed in various contexts.