CRN 46637 Math 536 001 Intro Diff Manifolds
Lecture Time TTh 12:30-13:45, Location CTLB-130
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: 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.
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;
Syllabus and Homework – Spring 2019
(please check after class as advanced postings of homework could change)
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.||
|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.|