## Diff Geo II WS 2009/10

 TU LV-Nr:3236 L 218 BMS Basic Course:Analysis and Geometry on Manifolds Lectures:Mon 12-14, Tue 14-16 in room MA 650 Tutorial:Wed 12-14 in room MA 650 Professor:John M. Sullivan, MA 318, tel. 314-29279 sullivan@math.tu-berlin.de, office hours TBA Assistant:Carsten Schultz cschultz@math.tu-berlin.de, office hours TBA Prerequisites:Analysis, elementary topology
 Modulprüfungen were offered on 3.3., 12.3., 9.4. and 21.5., in German or English. For a list of future dates and to schedule your oral exam, please see here. WWW: This syllabus is online at www.math.tu-berlin.de/~sullivan/L/09W/DG2/ The exercise sheets are available at carsten.codimi.de/diffgeo0910/ Course work:weekly homework assignments, one written test Test on 15 December 60 minutes (14:30-15:30); open book (you may use books and notes); Textbooks: Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd Ed, Academic Press Warner, Foundations of Differentiable Manifolds and Lie Groups, GTM 94, Springer Morgan, Riemannian Geometry: A Beginner's Guide, 2nd Ed, A K Peters Bishop and Goldberg, Tensor Analysis on Manifolds, Dover Milnor, Topology from the Differentiable Viewpoint, U P Virginia Spivak, Calculus on Manifolds, Benjamin/Cummings Sharpe, Differential Geometry, GTM 166, Springer Outline: This is a first course in manifolds and global analysis, which will present the basic tools for those interested in, or curious about, differential geometry or global analysis, or those who want to apply differentiial geometric methods in other areas such as PDE, topology, mathematical physics, and dynamical systems. The course will cover the following topics: Manifolds: Differentiable manifolds, implicit function theorem, rank theorem, tangent spaces, tangent bundles, vector bundles. Calculus on manifolds: Vector fields, flows, Lie bracket, Lie derivatives, Frobenius theorem. Differential forms: Differential forms, exterior calculus, orientability, Poincaré lemma, deRham complex. Integration theory: Stokes' theorem. Riemannian geometry: Riemannian metrics, distance, first variation and geodesics, Riemannian connection, curvature, connections on vector bundles.