CRN 46637 Math 536 001 Intro Diff
Manifolds
Lecture Time 9:009:50, SMLC 352
NOTE: class and office hours will be through Microsoft Office Teams at the usual times.
Instructor: Dimiter Vassilev Office :
SMLC, Office 326
Email: vassilev@unm.edu
Phone
Number:
505 277 2136
Office Hours: Monday & Friday 2pm  3pm. Feel free to stopby anytime when you have a quick question.
Final exam: check the official
Final
Examination Schedule.
Students having conflicts with this exam schedule must notify the appropriate instructor before TBD. Any student having more than three examinations scheduled in any one day may notify the instructor of the last examination listed. If notified before TBD, 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:
Lee J. M., Introduction to smooth manifolds (Springer, Graduate Texts in Mathematics) Corrections.
Other suggested textbook:
Morita S., Geometry of Differential Forms (Translations of Mathematical Monographs, Vol. 201).
Boothby, W., An Introduction to Differentiable Manifolds and Riemannian Geometry.
Tu, L., An Introduction to Manifolds.
Please
note the following guidelines for the course:
Description:
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: Makeup 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 2773506. In addition, they should see CATS
Counseling and Therapy Services;
Syllabus and Homework – Spring 2020
(please check after class as advanced postings of homework
could change)
Class Week 
Topics and Section 
Homework (due the 2nd class of the following week) in MS Teams 
1. Jan 20

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

HW1 
2. Jan 27  Examples using the manifold construction Lemma. Smooth partition of unity. Examples of diffeomorphsims; complex structure on 2sphere and complex projective line.  HW2 
3. Feb 3  Manifolds with boundary. The tangent space.  HW3 
4. Feb 10  The tangent bundle. The rank theorem; immersions and submersions; smooth covering maps  HW4 
5. Feb 17  Proper maps and covering maps. Immersed and (embedded) submanifolds  HW5 
6. Feb 24  Examples of submanifolds. Sard's theorem. Lie subgroups. Tangent space to a submanifold. Whitney's embedding theorem.  HW6 
7. March 2  Transversality. Locally trivial fiber bundles. Group actions.  HW7 
8. March 9  The quotient manifold theorem. Principle fiber bundles.  HW Homework 8 due March 25 
9. March 16  Spring Break  
10. March 23  Free and proper actions and PFB. Vector bundles.  HW9 
11. March 30  Vector bundels  constructions. Bundle homomorphisms and sections.  HW10 
12. Apr 6  Tensors. Differential forms. Characterization of tensor fields. Pullbacks of covariant tensors; diffeomorphsims and tensor fields. Midterm Exam (take home) 
HW11 
13. Apr 13  Derivations and contractions of tensor fields. The Lie derivative. Derivations and skewderivations of differential forms  interior product, exterior derivative, Lie derivative.  HW12 
14. Apr 20  de Rham cohomology, Poincare's lemma. Induced map on cohomologies, properties under homotopies. Cohomology and homotopic spaces. Volume form and Orientation. Integration of forms. Stokes' theorem.  HW13 
15. Apr 27  The general Stokes' theorem. Computing cohomology  MayerVietoris sequence, cohomology of spheres. Poincare lemma and cohomology with compact support.  HW14 
16. May 4  Flows of vector fields, complete vector fields  vector fields with compact suport, leftinvariant vector fields. Exponential of a Lie group, differential at the identity. Integrable distributions, Frobenius' theorem.  
Wednesday May 13  Final Exam 7:30am ‐ 9:30am, available in Teams at 7am. 