Diff Geo II
WS 2011

TU LV-Nr:

3236 L 218

BMS Basic Course:

Analysis and Geometry on Manifolds

Lectures:

Mondays 8:30-10:00 and 10:15-11:45 in room MA 313

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:

Felix Knöppel, MA 882, tel. 314-25783
knoeppel@math.tu-berlin.de, office hours TBA

Prerequisites:

Analysis, elementary topology

WWW:

Course information is online at page.math.tu-berlin.de/~sullivan/L/11W/DG2/

Course work:

weekly homework assignments, one written test on 7 December, oral final exam

Handouts

• Notes on exotic smooth structures

Homework assignments

• Homework 1, due 26 October
• Homework 2, due 2 November
• Homework 3, due 9 November – note corrected coordinates for poles
• Homework 4, due 16 November
• Homework 5, due 23 November
• Homework 6, due 30 November
• Homework 7, review for the test on 7 Dec – not to be turned in
• Homework 8, due 4 January
• Homework 9, due 11 January
• Homework 10, due 18 January
• Homework 11, due 25 January
• Homework 12, due 1 February
• Homework 13, due 8 February
• Homework 14, due 15 February

Textbooks:

Kühnel, Differential Geometry / Differentialgeometrie, AMS / Vieweg
Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd Ed, Academic Press
Bröcker and Jänich, Intro to Differential Topology / Einführung in die Differentialtopologie, CUP / Springer
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.