Differential Algebras with Dense Singularities on Manifolds
Abstract
Recently the spacetime foam differential algebras of generalized functions with dense singularities were introduced, motivated by the so called spacetime 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 CauchyKovalevskaia theorem, de Rham cohomology in abstract differential geometry, and so on. So far the spacetime 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 spacetime foam algebras, their extension to manifolds proceeds in a surprisingly easy and natural way. It is also shown that the spacetime 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606358
 Bibcode:
 2006math......6358R
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Mathematical Physics;
 Mathematical Physics