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Front page > Course > SIF5034
MA3402 Analysis on Manifolds
Fall 2006
About the course | Lecturer | Textbook and Curriculum | Exam | Lectures | Practice session | Exercises | About this page
| About the course |
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|---|---|
| Lecturer | Professor Eldar Straume |
| Textbook: Spivak's book Supplementary notes: 1) Tensor Calculus- a brief overview 2) tensorfields 3) Handwritten lecture notes from Chap. 8 in Spivak |
1. M.Spivak: A Comprehensive Introduction to Differential Geometry, Volume I , Publish or Perish, Inc, Houston,Texas 1999. 2 Tensor Calculus- a brief overview. A somewhat popularized form, which may be useful for you. Most books on differential topology cover more or less our topics in the curriculum, see e.g. the web page of the spring 2004 course TMA4190. In particular, it is useful to have the text book of TMA4190: 3. James R.Munkres : Analysis on manifolds. (Try internet bookstores, such as Amazon) Chap 6-Chap.9, Chap.6 is particularly nice for our course. 4. Per Holm: Differensialformer og Mangfoldigheter (in Norwegian) Some of the chapters discuss vector bundles, differential forms etc. Curriculum (from Spivak): not updated yet Chap. 4 : Chap. 5 : Chap. 7 : (not Frobenious integrability theorem) Chap. 8 : Integration (except the proofs of Theorem 9,10,11) a Chap 9: Riemannian metrics, only basic definitions page, .311-315. Chap. 11: Excursion in the realm of algebraic topology: side 419-425, s.430- 431 . The Exercise sets may also be useful for the final exam, and they give you some experience. In addition, the student need some basic knowledge of differential topology, for example from Spivak Chap.1-3. About manifolds and vector bundles. |
| About
the exam. Time : |
Final Grades |
| Lectures Plans and progress Remarks |
Thursday 12.15-14.00 Room R59 (Realfag building) Wednesday 13.15-15, Room K25 (The Friday time has been changed to Wednesday) Week 34 : We start with a brief review of basic material about manifolds and vector bundles. See Spivak, chap.1 -chap.3 Week 35: Read about the tangent bundle and (the dual) cotangent bundle. We have discussed the tangent bundle and vector bundles in general. Week 36 . We continue with Chapter 4 and will also introduce the tensorbundles of a smooth manifold . These are vector bundles , and the simplest ones are the tangent bundle and the cotangent bundle. .Their sections are called vector fields and covector fields (or 1-forms) respectively. From this one can construct all possible tensorfields; they are just the cross sections of the appropriate tensor bundle. In the Supplementary notes you will find the local expressions for such tensor fields etc. tentative plan: Week 37-38 : After you have got some overview of tensors and tensorfields in chap.4, you should start going through chap. 5 about the connection between vector fields and diff. equations. We shall go through the main themes there, but we shall not go through the existence and uniqueness proofs since they are time consuming. Week 39/40 : Chapter 5 : vector fields, dynamical systems etc. Week 40 : We also start with chap. 7 : differential forms Week 43: We complete Chap.7 Week 44-46: Chapter 8 : Integration theory Week 47 : The remaining topics: from Chap. 8, Chap.11 (Mayer-Vietoris) and a little from Chap.9 (Riemannian metric) |
| Homework/Exercises Exercise set 1 Exercise set 2 Exercise set 3 Midsemester project Exercise set 4 Exercise set 5 |
Week 35 :Exercise
set 1 Week 37 : Exercise set 2 (same as in 2005) Week 39/40 : Spivak Chap. 5.11a,b, 5.12, 5.13 Week 41 : Exercise set 3 Week 42 : Midsemester project Week 43 : Spivak Chap.7: 13, 14a,b, 27 Week 44: Exercise set 4. Week 45-46 : Spivak Chap.8 : 3, 4,7,13,14,17 (see also the note "Notes/hints to Chap.8" ) Week 47 : Exercise set 5. This is an exercise concerning Riemannian metric |
| Office hours | Until further notice : any time |
| About this page |
The address of this page is
http://www.math.ntnu.no/~eldars/MA3402eng06.htm
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