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There will be a two-semester cycle on formal integrability of nonlinear systems of partial differential equations (leading to involutive bases), given by Markus Rosenkranz and Georg Regensburger. This semester we start out by introducing its most essential tool: differentiable manifolds. Due to the shortage of time (introductions to manifolds are typically given in a four-hour course plus one hour exercises!) and due to the (pleasently) intensive interaction with the audience, we have only covered the very basic notions. We plan to continue this, partly in the next semester and maybe also in the upcoming semesters.
Here is the table of contents of the lecture notes:
0 Preliminaries 0.1 Vector Spaces 0.2 Change of Bases 0.3 Maps in Toplogical Spaces 0.4 Maps in Vector Spaces
1 The Category of Manifolds 1.1 Embedded Manifolds 1.1.1 Differential Calculus in Vector Spaces 1.1.2 Manifolds in Vector Spaces 1.1.3 From Ambient to Abstract Charts 1.2 Abstract Manifolds 1.2.1 The Chart Topology 1.2.2 Differentiable Structures 1.2.3 Manifolds as Patchwork 1.2.4 The Definition of Manifold
2 The Tangent Space 2.1 Cotangent and Tangent Vectors 2.1.1 Abstract Setting 2.1.2 Representation through Components 2.1.3 Co- and Contravariance 2.1.4 Coderivations and Derivations 2.2 The Differential 2.2.1 Abstract Setting 2.2.2 Representation through Components 2.2.3 The Special Case of Vector Spaces 2.2.4 Co- and Contravariance Revisited 2.2.5 The Rank Theorem 2.3 Tensor Fields on Manifolds 2.3.1 Fiber Bundles 2.3.2 Vector Bundles 2.3.3 The Tensor Bundles 2.3.4 Pullback and Pushforward 2.3.5 Various Tensor-like Quantities
The latest version (as of 30 Jan 2007 / 13:14) of the lecture notes is here. This is supposed to be the final version, but of course there will be corrections and additions later, as time allows. Questions, comments and constructive criticism (including typos that you spotted!) are very welcome: Either write an email to the above addresses or approach us directly.