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AuthorSchiekel, Bernharddc.contributor.author
Date of accession2019-08-01T11:08:51Zdc.date.accessioned
Available in OPARU since2019-08-01T11:08:51Zdc.date.available
Date of first publication2019-07-22dc.date.issued
AbstractThis publication is a propaedeutic monograph about Gauss-Bonnet theorems and Atiyah-Singer indextheorems (ASI). Prerequisites are advanced undergraduate level in mathematical physics and some interest and time. Topics are the development of the notions of curvature and topological invariants through the ages and their application to physics. The beginnings in this field were given by Euklid, Archimedes, Harriot, Girard, Newton, Frenet-Serret. Next we come to Euler with his topological invariant 'Euler charcteristic', Gauss & Bonnet with their theory of 2-dimensional surfaces, Riemann with his 'Differential Geometry' and Einstein-Cartan with gravity based on curvature + torsion (ECSK)'. Then the story goes on with 'Homotopy', 'Simplicial Homology', 'Singular Homology', 'Cohomology', 'Hodge theory', 'Lie-Groups', 'Fibre Bundles', 'Gauge theory', 'Characteristic Classes' until a pathintegral-proof of the ASI for the chiral euklidean Dirac-Operator á la Witten et al. The extended 2. edition adds 'Clifford-Algebras', 'Nieh-Yan characteristic class', '3+1 Gravity', 'Palatini-Holst- & Plebański-Gravity' and the 'BF-model'. We owe all this (and much more) to Élie Cartan, Poincaré, Einstein, Hopf, Brower, de Rham, Hodge, Lie, Lorentz, Dirac, Ehresmann, Yang & Mills, Chern, Weil, Pontrjagin, Todd, Hirzebruch, Atiyah & Singer and Witten. Our motto is in the words of Spivak: "All the way with Gauss-Bonnet". The language is German. Due to the didactical intention of this introduction all proofs are worked out in details. So, enjoy yourself! :-)dc.description.abstract
Languagededc.language.iso
PublisherUniversität Ulmdc.publisher
Earlier versionhttp://dx.doi.org/10.18725/OPARU-4419dc.relation.isversionof
LicenseStandarddc.rights
Link to license texthttps://oparu.uni-ulm.de/xmlui/license_v3dc.rights.uri
KeywordAtiyah-Singer Theoremdc.subject
Dewey Decimal GroupDDC 530 / Physicsdc.subject.ddc
LCSHGauss-Bonnet theoremdc.subject.lcsh
LCSHMathematical physicsdc.subject.lcsh
LCSHHomotopy theorydc.subject.lcsh
LCSHPhysicsdc.subject.lcsh
LCSHTopologydc.subject.lcsh
LCSHGeometry, differentialdc.subject.lcsh
TitleKrümmungen und Indexsätze - auf den Spuren von Gauß-Bonnet, Cartan, Atiyah-Singer und Witten. Eine Einführung in Geometrie und Topologie für Physiker.dc.title
Resource typeBuchdc.type
DOIhttp://dx.doi.org/10.18725/OPARU-17162dc.identifier.doi
PPN1677322136dc.identifier.ppn
URNhttp://nbn-resolving.de/urn:nbn:de:bsz:289-oparu-17219-3dc.identifier.urn
GNDHomotopiedc.subject.gnd
GNDDifferentialgeometriedc.subject.gnd
FacultyFakultät für Naturwissenschaftenuulm.affiliationGeneral
EditionErweiterte 2. Auflageuulm.edition
DCMI TypeTextuulm.typeDCMI
CategoryPublikationenuulm.category
URL of original publicationhttps://www.mb-schiekel.de/gauss-bonnet-atiyah-singer.pdfdc.relation1.url
Bibliographyuulmuulm.bibliographie


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