CRN 46637 Math 536 001 Intro Diff Manifolds

Lecture Time TTh 12:30-13:45, Location CTLB-130

 

                                                                 

Instructor: Dimiter Vassilev     Office :  SMLC, Office 326  Email: vassilev@unm.edu  Phone Number: 505 277 2136

 

Office Hours: Tuesday 9:30-10:30am, Thursday 4:30-5:30pm. Feel free to stop-by anytime when you have a quick question.

 

Final exam: Thursday, May 9, 10:00–12:00 (usual room) Please double check the official Final Examination Schedule.

Students having conflicts with this exam schedule must notify the appropriate instructor before Friday, March 29, 2019. Any student having more than three examinations scheduled in any one day may notify the instructor of the last examination listed. If notified before Friday, March 29, 2019, the instructor shall make arrangements to give a special examination. Changes in this examination schedule are not permitted except by formal approval of the instructor’s College Dean.  

 

Textbook:

 

Other suggested textbook:

Please note the following guidelines for the course:

 

Description: Concept of a manifold, differential structures, vector bundles, tangent and cotangent bundles, embedding, immersions and submersions, transversality, Stokes' theorem.

 

Prerequisite: Math 511.


Grades: The final grade will be determined by homework (100 points), one midterm exam (100 pts) and a final exam (200 points). Not all homework problems will be graded. You will get credit if you turn in the statements of all the homework problems written in the correct order. The Final Exam score will replace the midterm score if the latter is lower than the Final Exam score.  All grades will be posted on UNM Learn.

 

Homework:  You can work together on the homework, but you do need to write your own solutions in your own words. To help the grader, please write your solutions up neatly and clearly and staple the sheets.   Each homework has a total of 20 points.  The Extra Problems (if any) are not due. Not all homework problems will be graded.

 

Missed Exams:  Make-up exams can be arranged for exams missed with a VALID excuse (illness, family emergency, active participation in scholarly or athletic activities), and ONLY if prior notice is given. 

 

Disability Statement: We will accommodate students with documented disabilities. During the first two weeks of the semester, those students should inform the instructor of their particular needs and they should also contact Accessibility Services in Mesa Vista Hall, Room 2021, phone 277-3506.  In addition, they should see CATS- Counseling and Therapy Services; Student Health Center (277-4537). (They can help if you suffer from exam anxiety).    

 

 

 

 Syllabus and Homework – Spring 2019 

(please check after class as advanced postings of homework could change)

 

 

Class Week

Topics and Section

Homework (due the 1st class of the following week)
1. Jan 14

 

 1. Topological and smooth manifolds - definitions and examples, paracompactness, partition of unity, smooth structures defined by an atlas.

HW1:

 
2. Jan 21 Manifolds with boundary. 2. Smooth Maps. Locally trivial fibrations. HW2:
3. Jan 28 Diffeomorphisms, smooth partitions of unity. 3. Tangent vectors and the tangent space at a point. HW3:
4. Feb 4 The tangent bundle. 4. The rank theorem; immersions and submersions. HW4:
5. Feb 11 Smooth covering maps. Embeddings. Lie groups. HW5:
6. Feb 18 Feb. 19: snow day. Submanifolds. HW6:
7. Feb 25 Submanifolds-regular level set, uniqueness of smooth structure; O(n), SL(n). Tangent space to a submanifold. Defining function of the boundary and regular domains. The Whitney embeding theorem. Transversality. HW7:
8. March 4 Fiber product theorem for transversal maps. 21. Group actions and the quotient manifold theorem. Thursday, March 7, Midterm Exam. Solutions. HW8:(due Tusday, March 19)
9. March 11 March 10-17 Spring Break  
10. March 18 Free and proper group actions and the manifold M/G. Principle fiber bundles. HW9:
11. March 25 Vector bundles. Tautological line bundle over the projective space. Constructing new bundles from old. Integrable/ non-integrable subbundles of the tangent bundle.HW10:
12. Apr 1 The tensor bundle and tensor fields, characterization of (0,q)-tensors. Straightening of a vector field near a regular point. HW11:
13. Apr 8 Proof of Frobenius' theorem. Integral curves and the flow of a vector field. Complete vector fields, 1-parameter group of diffeomorphism, infinitesimal generator. The Lie algebra of a Lie group, 1-parameter subgroups of a Lie group. HW12:
14. Apr 15 The Lie derivative of a vector field- commuting flows. Derivations of the tensor algebra. The Lie derivative as a derivation of the tensor algebra. Derivations and skew-derivations of forms, the exterior derivative. HW13:
15. Apr 22 The interior multiplication. Cartan's formula. Closed and exact forms, de Rham cohomology and homotopies. Integration of differential forms and Stokes' theorem. HW14:
16. Apr 29 Stokes' theorem for a manifold with boundary. Mayer-Vietoris sequence- cohomology of spheres. The degree of a map, index of a vector field and Hopf's theorem. Frobenius' theorem in the language of forms (differential ideals). Review group actions - homogeneous spaces.  
Thursday May 9 Final Exam 10:00 a.m.‐12:00 p.m. in the usual room.