  Recently the space-time foam differential algebras of generalized functions
with dense singularities were introduced, motivated by the so called space-time
foam structures in General Relativity with dense singularities, and by Quantum
Gravity. A variety of applications of these algebras has been presented, among
them, a global Cauchy-Kovalevskaia theorem, de Rham cohomology in abstract
differential geometry, and so on. So far the space-time foam algebras have only
been constructed on Euclidean spaces. In this paper, owing to their relevance
in General Relativity among others, the construction of these algebras is
extended to arbitrary finite dimensional smooth manifolds. Since these algebras
contain the Schwartz distributions, the extension of their construction to
manifolds also solves the long outstanding problem of defining distributions on
manifolds, and doing so in ways compatible with nonlinear operations. Earlier,
similar attempts were made in the literature with respect to the extension of
the Colombeau algebras to manifolds, algebras which also contain the
distributions. These attempts have encountered significant technical
difficulties, owing to the growth condition type limitations the elements of
Colombeau algebras have to satisfy near singularities. Since in this paper no
any type of such or other growth conditions are required in the construction of
space-time foam algebras, their extension to manifolds proceeds in a
surprisingly easy and natural way. It is also shown that the space-time foam
algebras form a fine and flabby sheaf, properties which are important in
securing a considerably large class of singularities which generalized
functions can handle.
