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Curvature in Differential Geometry and General Relativity SemesterSoSe 2019 Lecturer Felix Finster, Marco Oppio Type of course (Veranstaltungsart) Seminar Contents The purpose of this seminar is to introduce the concept of curvature of a manifold and to lightly touch one of the perhaps most fascinating examples: Einstein’s spacetime. In order to get familiar with the topic, the seminar starts with an elementary introduction to regular surfaces in Euclidean space. Concepts like tangent plane, orientation, first and second fundamental forms, mean and Gaussian curvature and minimal surfaces are discussed. Finally, the Riemann tensor is introduced and Gauss’ Theorema Egregium is proved. As a conclusion of this section and as a step towards the abstraction of the second part, general tools like Riemann metric, geodesics and parallel transport are discussed in the special case of regular surfaces. In the second part, the attention is drawn to abstract Riemannian manifolds. Concepts like a tangent space, tensor field, (pseudo-)Riemannian metric, covariant derivative, geodesic and the Riemann tensor are reviewed in full generality. This comes in preparation of the last section of this seminar, whose goal is to give an introduction to the basics of general relativity, with a focus on the mathematical formulation of the underlying physical concepts. After a brief discussion on the main principles upon which the special theory of relativity is constructed – the constancy of the speed of light over all – the seminar aims at introducing the generalizations proper to general relativity: mainly the principle of general covariance and the equivalence principle. As ultimate goal, the Einstein field equations are motivated and the perhaps most remarkable solution is discussed: the Schwarzschild black hole spacetime. The seminar is addressed and accessible to mathematics students (even without physical background). The last talks on General Relativity should be given preferably by students with physical interests. Literature Christian Bär: Elementare Differentialgeometrie Manfredo do Carmo: Differentialgeometrie von Kurven und Flächen Robert Wald: General Relativity Barrett O’Neill: Semi-Riemannian Geometry more literature will be given in the seminar Recommended previous knowledge Analysis I-III Time/Date Wednesday 8:30-10:00 Location M 103 Course homepage https://www.uni-regensburg.de/mathematik/mathematik-1/index.html (Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern) Registration
BSem, LA-GySem ECTS Siehe Modulkatalog. MV und Nebenfach: 4,5 LP bei Studienbeginn ab WS 15/16, 6 LP bei Studienbeginn vor WS 15/16 |