Krümmungen und Indexsätze - auf den Spuren von Gauß-Bonnet, Cartan, Atiyah-Singer und Witten. Eine Einführung in Geometrie und Topologie für Physiker.

Abstract

This 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! :-)