MATH 439001 55949 ST: Intro to Diff Manifold & Introduction to Differentiable Manifolds  51950  MATH 536  001
Class time 11:00 am12:15 pm Location SMLC 352.
Instructor:
Dimiter Vassilev Office :
SMLC, Office 326
Email: vassilev@unm.edu
Phone
Number:
505 277 2136
Required textbook: An Introduction to Differential Manifolds (2015), Authors: Jacques Lafontaine
http://link.springer.com/book/10.1007%2F9783319207353 (available free for download on the UNM network)
Topic of the course: Chapters 16 of Lafontaine’s book. Description: Concept of a manifold, differential structures, tangent and cotangent bundles, embedding, immersions and submersions, transversality, differential forms and integration, Stokes' theorem, Lie groups.
Other Textbooks you might find useful:
Lee J. M., Introduction to smooth manifolds (Springer, Graduate Texts in Mathematics)
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:
Prerequisites: Linear Algebra (MATH321), Calculus III (MATH264) and at least two of the following courses: Topology (MATH431), Analysis (MATH401) , Analysis (MATH402).
Grades: The final grade will be determined by the homework (100 pts), two
midterm exams (100 pts) and
a final exam (200 points). All grades will
be posted on
UNM Learn.
Homework: You can work together on the homework, but you do need to write up 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 5 problems (4 pts each) for a total of 20 points. The Extra Credit Problems are not due and are truly extra credit (8 pts each). It is best if you work and discuss the problems together and with me as questions arise. The highest 10 homeworks will be counted.
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;
Homework is hosted on UNM Learn
(please check after class as advanced postings of homework
could change)
Week
of 
Topics
Covered 
Homework due Thursday following week at the beginning of class. 
Jan. 18 
1. Differentials, rank theorem  diffeomorphisms. 
HW1 due Jan. 28 
Jan. 25 
2. The rank theorem, immersions, submersions. Smooth submanifolds of Rn. 
HW2 due Feb. 4 
Feb. 1 
3. Level sets, graphs and parametrizations. Examples. 
HW3 due Feb. 11 
Feb. 8 
4. The tangent space. Sets of measure zero and their images under smooth maps. Critical points and values.

HW4 due Feb. 18 
Feb. 15 
5. Sard's theorem. Transversality. Topological and smooth manifolds. 
HW5 due Feb. 25 
Feb. 22 
6. Construction of Manifolds. Smooth Maps  nonequivalent smooth structures, diffeomorphic smooth structures. Examples  Projective Spaces. 
HW6 due Mar. 3 
Feb. 29 
7. Fibrations, the Hopf fibration. The Fundamental Theorem of Algebra. 
HW7 due Mar. 10, Feb. 29 Homework Session SMLC 352 9:4510:45 
Mar. 7 
8. Tangent space. Local diffeomorphisms, immersion, submersion, submanifolds. Midterm Exam 1 
HW8 due Mar. 31 
Mar. 14 
Spring
Break 

Mar. 21 


Mar. 28 

HW9 due Apr. 7 makeup class SMLC 352 9:3010:45 
Apr. 4 

HW10 due Apr. 14 Apr. 4 Homework Session SMLC 352 9:4510:45 Apr. 8 makeup class SMLC 352, 9:3010:45 
Apr. 11 

HW11 due Apr. 21 
Apr. 18 
12. T Midterm Exam 2

HW12 due Apr. 28 Monday Homework Session SMLC 352 9:4510:45. Makeup classes: Wednesday & Friday 9:45am10:55am. 
Apr. 25 
13. Integration of differential forms. Manifolds with boundary. Stokes theorem. DeRham cohomology. Degree of a map.
14. Lie groups  left invariant vector fields, the Lie algebra, exponential mapping, Lie subgroups. Group actions. Quotient manifolds and homogeneous spaces. 
HW13 due May 5 Makeup class: Monday 9:45am10:45am No class on Tuesday Makeup class: Wednesday 9:45am10:45am Makeup class: Friday 9:00am10:15am 
May 2 

No classes, but turn in the due homework on Thursday as usual and collect your old homework. 
May 9 Finals week 
Final Exam May 10, 12:30pm  2:30pm 
