Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

ALGEBRAIC FOUNDATIONS OF THE THEORY OF DIFFERENTIAL SPACES

ALGEBRAIC FOUNDATIONS OF THE THEORY OF DIFFERENTIAL SPACES DEMONSTRATIO MATHEMATICAVol. XXIVNo 3-41991Mlchal HellerALGEBRAIC FOUNDATIONS OF THE THEORY OF DIFFERENTIAL SPACESIn physics there isgeometricmodelsofanurgentphysicalnon-smooth" generalizationsconcept. Thetheoryofnecessityphenomenaofthetoonbase"sufficientlydifferentiabledifferentialspacessomemanifoldmightprovidephysics with such a possibility. Algebraic foundations of thistheory are discussed.DifferentialSikorski turns out tobealgebraicconceptofaspace"geometricringedspace,inthesenserefinement"anditofofthenaturallygeneralizes the real manifold concept. However, itprovesbemanifolds.inadequatetodealwithcomplexanalyticMostow's theory of differential spaces is a geometricof the theory of structured spaces(essentially,tonaturallymanifoldonegeneralizemusttheconceptsuitablyadaptofversionsheavesgerms of functions on a topological space). It isshowncomplexMostow'stoofthatanalyticconceptofdifferential space.0. Introduct ionThereisawidespreadconvictiongeometry must be constrained to"smooth enough", in practice withdealthatwithspacesspaceswhichdiffeomorphic to a Euclidean space. We havedifferentialthatarelearnedarelocallytolivewith this view, but in fact it is a cumbersome constraint. Theworld around us is far from being "smooth enough", andwould like to haveitsshouldlookardentlydifferentiallyforgeometric"sufficientlyifwemodel,wenon-smooth"350M. Hellergeneralizations of thedifferentiableturns out that for severalyearsamong mathematicians quite amanifoldtherefewaresuchinstance [1] - [9], see also [10] andconcept.incirculationgeneralizationsoriginaltherein), although some of them in a seminalItworksstate(forquotedoftheirdevelopment. In almost all of these works it is preciselytheassumption of the local resemblance to a Euclidean spacethatis rejected, and the structure thus obtained is usually calleda differentialspace.W h e n one starts to applytheseto physical problems, they seemed to work wellmethods([12] -[13]);especially encouraging results have been obtained in t h e fieldof classical space-time singularitiescircumstancesthequestionis([14] - [15]).unavoidable:InWhydosuchthesemethods work? Our hitherto views on differential geometry h a v eturned out to be strongly biased. W h i c h is http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

ALGEBRAIC FOUNDATIONS OF THE THEORY OF DIFFERENTIAL SPACES

Heller, Mlchal
Demonstratio Mathematica , Volume 24 (3-4): 16 – Jul 1, 1991
Download PDF
16 pages

Loading next page...
 
/lp/de-gruyter/algebraic-foundations-of-the-theory-of-differential-spaces-PY4q8QFtXZ

References

References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.

Publisher
de Gruyter
Copyright
© by Mlchal Heller
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1991-3-403
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. XXIVNo 3-41991Mlchal HellerALGEBRAIC FOUNDATIONS OF THE THEORY OF DIFFERENTIAL SPACESIn physics there isgeometricmodelsofanurgentphysicalnon-smooth" generalizationsconcept. Thetheoryofnecessityphenomenaofthetoonbase"sufficientlydifferentiabledifferentialspacessomemanifoldmightprovidephysics with such a possibility. Algebraic foundations of thistheory are discussed.DifferentialSikorski turns out tobealgebraicconceptofaspace"geometricringedspace,inthesenserefinement"anditofofthenaturallygeneralizes the real manifold concept. However, itprovesbemanifolds.inadequatetodealwithcomplexanalyticMostow's theory of differential spaces is a geometricof the theory of structured spaces(essentially,tonaturallymanifoldonegeneralizemusttheconceptsuitablyadaptofversionsheavesgerms of functions on a topological space). It isshowncomplexMostow'stoofthatanalyticconceptofdifferential space.0. Introduct ionThereisawidespreadconvictiongeometry must be constrained to"smooth enough", in practice withdealthatwithspacesspaceswhichdiffeomorphic to a Euclidean space. We havedifferentialthatarelearnedarelocallytolivewith this view, but in fact it is a cumbersome constraint. Theworld around us is far from being "smooth enough", andwould like to haveitsshouldlookardentlydifferentiallyforgeometric"sufficientlyifwemodel,wenon-smooth"350M. Hellergeneralizations of thedifferentiableturns out that for severalyearsamong mathematicians quite amanifoldtherefewaresuchinstance [1] - [9], see also [10] andconcept.incirculationgeneralizationsoriginaltherein), although some of them in a seminalItworksstate(forquotedoftheirdevelopment. In almost all of these works it is preciselytheassumption of the local resemblance to a Euclidean spacethatis rejected, and the structure thus obtained is usually calleda differentialspace.W h e n one starts to applytheseto physical problems, they seemed to work wellmethods([12] -[13]);especially encouraging results have been obtained in t h e fieldof classical space-time singularitiescircumstancesthequestionis([14] - [15]).unavoidable:InWhydosuchthesemethods work? Our hitherto views on differential geometry h a v eturned out to be strongly biased. W h i c h is

Journal

Demonstratio Mathematica – de Gruyter

Published: Jul 1, 1991

There are no references for this article.