On the Differential Geometry of Some Classes of Infinite Dimensional Manifolds

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 $X$.