The differential space concept : a generalization of the manifold concept

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dc.contributor.advisorCherenack, P Fen_ZA
dc.contributor.authorNtumba, Patrice Punguen_ZA
dc.date.accessioned2014-11-10T08:59:30Z
dc.date.available2014-11-10T08:59:30Z
dc.date.issued1997en_ZA
dc.descriptionBibliography: leaves 87-88.en_ZA
dc.description.abstractIt 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.en_ZA
dc.identifier.apacitationNtumba, P. P. (1997). <i>The differential space concept : a generalization of the manifold concept</i>. (Thesis). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/9484en_ZA
dc.identifier.chicagocitationNtumba, Patrice Pungu. <i>"The differential space concept : a generalization of the manifold concept."</i> Thesis., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 1997. http://hdl.handle.net/11427/9484en_ZA
dc.identifier.citationNtumba, P. 1997. The differential space concept : a generalization of the manifold concept. University of Cape Town.en_ZA
dc.identifier.ris TY - Thesis / Dissertation AU - Ntumba, Patrice Pungu AB - 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. DA - 1997 DB - OpenUCT DP - University of Cape Town LK - https://open.uct.ac.za PB - University of Cape Town PY - 1997 T1 - The differential space concept : a generalization of the manifold concept TI - The differential space concept : a generalization of the manifold concept UR - http://hdl.handle.net/11427/9484 ER - en_ZA
dc.identifier.urihttp://hdl.handle.net/11427/9484
dc.identifier.vancouvercitationNtumba 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/11427/9484en_ZA
dc.language.isoengen_ZA
dc.publisher.departmentDepartment of Mathematics and Applied Mathematicsen_ZA
dc.publisher.facultyFaculty of Scienceen_ZA
dc.publisher.institutionUniversity of Cape Town
dc.subject.otherMathematicsen_ZA
dc.titleThe differential space concept : a generalization of the manifold concepten_ZA
dc.typeMaster Thesis
dc.type.qualificationlevelMasters
dc.type.qualificationnameMScen_ZA
uct.type.filetypeText
uct.type.filetypeImage
uct.type.publicationResearchen_ZA
uct.type.resourceThesisen_ZA
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