It is well known that the theory of smooth manifolds is incapable in dealing with the classical singularity problem in relativistic cosmology and relativistic astrophysics. To overcome this problem, attempts have been made to obtain a more general and geometrically manageable concept than the traditional manifold concept. It is to this end that Aronszajn and Marshall developed the theory of the so-called sub-cartesian spaces that essentially are manifolds with "singularities" such as piecewise manifolds and quasianalytic sets of Rn. In the spirit of this generalization, Sikorski proposed the so-called differential space (or d-space, for short) by dropping the axiom forcing the manifold to be locally diffeomorphic to the Euclidean space of some dimension. Any subset of Rn is a d-space, and there are many d-spaces which cannot be embedded in any Euclidean differential manifold. This makes the differential space concept a suitable tool to deal with the classical singularity problem. One should notice that space-time with its singular boundary is no longer a differentiable manifold, but it can be viewed as a differential space. Singularities (at least regular and some quasi-regular singularities) need not be considered as belonging to "singular boundaries" of space-time, but can be regarded as "internal domains" of a corresponding differential space . The aim of this work is to show the applicability of differential space theory in relativistic cosmology. As we will see, the notion of differential space enables us to investigate problems in differential geometry where differentiable manifolds do not suffice; for instance a geometrical analysis of quasi-regular singularities is possible within this framework. In this respect, we devote some time to the quasi-regular singularities of both the cosmic string and closed Friedman world. According to the classification scheme developed by Ellis and Schmidt, quasi-regular singularities are defined as those points of space-time through which no space-time extension is possible although the local geometry is well behaved as one approaches the singularity point. An observer approaching such a singularity has no warning until his history abruptly comes to an end.
Reference:
Ntumba, P. 1997. The differential space concept : a generalization of the manifold concept. University of Cape Town.
Ntumba, P. P. (1997). The differential space concept : a generalization of the manifold concept. (Thesis). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/9484
Ntumba, Patrice Pungu. "The differential space concept : a generalization of the manifold concept." Thesis., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 1997. http://hdl.handle.net/11427/9484
Ntumba PP. The differential space concept : a generalization of the manifold concept. [Thesis]. University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 1997 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/9484