On the Differential Geometry of Some Classes of Infinite Dimensional Manifolds

TL;DR

This paper develops a general algebraic framework for lifted differential geometry on infinite-dimensional spaces related to a manifold, enabling analysis without local coordinates, with applications to measures, submanifolds, and tilings.

Contribution

It introduces a coordinate-free algebraic framework for lifted geometry applicable to various infinite-dimensional spaces associated with a manifold.

Findings

Constructed a general algebraic framework for lifted geometry.

Applied lifted geometry to spaces of measures, mappings, submanifolds, and tilings.

Connected Stokes' theorem and boundary differentiability in lifted geometry.

Abstract

Albeverio, Kondratiev, and R\"{o}ckner have introduced a type of differential geometry, which we call lifted geometry, for the configuration space $\Gamma_X$ of any manifold $X$. The name comes from the fact that various elements of the geometry of $\Gamma_X$ are constructed via lifting of the corresponding elements of the geometry of $X$. In this note, we construct a general algebraic framework for lifted geometry which can be applied to various ``infinite dimensional spaces'' associated to $X$. In order to define a lifted geometry for a ``space'', one dose not need any topology or local coordinate system on the space. As example and application, lifted geometry for spaces of Radon measures on $X$, mappings into $X$, embedded submanifolds of $X$, and tilings on $X$, are considered. The gradient operator in the lifted geometry of Radon measures is considered. Also, the construction of a natural Dirichlet form associated to a Random measure is discussed. It is shown that Stokes' Theorem appears as ``differentiability'' of ``boundary operator'' in the lifted geometry of spaces of submanifolds. It is shown that (generalized) action functionals associated with Lagrangian densities on $X$ form the algebra of smooth functions in a specific lifted geometry for the path-space of $X$.