KTH    Matematik

Differential Geometry, Manifolds and Differential Forms, Fall 2011 Course at advanced level (course number KTH: SF2722, SU: MM8022) and graduate level (course number KTH: FSF3674, SU: XXXX), 7.5 credits. General information about courses given on both advanced and graduate level can be found here. Teacher: Mattias Dahl, dahl@math.kth.se. Literature: Jeffrey M. Lee, Manifolds and Differential Geometry, (AMS, Google, amazon). A list of misprints is here, please mail if you find more! Time and place: Lectures are Mondays 10.15-12.00 in seminar room 3721 at KTH Mathematics, The first meeting is Monday 5/9 and the course will go on until Monday 19/12. Lectures: Comments to the lectures, reading instructions and recommended exercises can be found here. Description: The central objects in modern differential geometry are differentiable manifolds. In this course we will study differentiable manifolds and see how they are used to define concepts from analysis in a coordinate-independent way. We will see how to define tensors and differential forms and how to formulate the fundamental theorem of calculus in geometric way as Stokes' theorem. In an introduction to (semi-)Riemannian geometry we will see how curvature is described. Ideas and methods from differential geometry are fundamental in modern physical theories. Content: The content of the advanced level course are the following chapters from the course book. 1. Differentiable Manifolds 2. The Tangent Structure 3. Immersions and Submersions 4. Curves and Hypersurfaces in Euclidean space 7. Tensors 8. Differential forms 9. Integration and Stokes' Theorem 13. Riemannian and Semi-Riemannian Geometry. For the graduate level course the students should read further chapters from the book and solve additional homework problems. According to your choice you will study some (but not all) of the following. 5. Lie Groups 6. Fiber Bundles 11. Distributions and Frobenius' Theorem further material from chapters 9 and 13. Prerequisites: SF2713, Foundations of Analysis, or corresponding background. Also recommended is SF2729, Groups and Rings, or equivalent. A good knowledge of calculus of several variables including the inverse and implicit function theorems is an important prerequisite. Examination: The examination will be in the form of homework problems followed by an oral examination. For grades C-E you only have to solve the homework problems. For grades A-B and for graduate students you must also do the oral examination. Homework problems: For the advanced and graduate level course, Homework 1, chapters 1, 2, Homework 2, chapters 3, 4, (NEW VERSION AGAIN, there were errors in Problems 1, 2), Homework 3, chapters 7, 8, 9, Homework 4, chapter 13, (NEW VERSION, error in problem 2). Additional homework for the graduate level course, hand in solutions to two of the following, Graduate homework 1, chapter 5, Graduate homework 2, chapter 6, Graduate homework 3, chapter 11, Graduate homework 4, chapter 13. There is no particular deadline for the homeworks during this semester (but later there might be one). Oral examination: When you have handed in all the homework exercises please contact me to arrange a time for the oral examination. The day before the examination day I will mail you 2-3 problems from the weekly "recommended exercises" in the reading instructions. In the examination you must then present clear and complete solutions to these problems.

Sidansvarig: Mattias Dahl