Math 761 (Fall 2017)
Differentiable Manifolds
Differentiable Manifolds
Announcements
There will be a make-up lecture on Nov. 16 from 7pm to 10pm in B139 Van Vleck.
Lectures on Nov. 3 and 7 will be rescheduled to a later time (TBA).
Lecture on Oct. 11 will be rescheduled to a later time (TBA) of the semester.
Welcome to the class! The first lecture is at 9:55am on Wednesday, Sep. 6, in Van Vleck B235.
Lectures
Time: 9:55am-10:45am, on Mondays, Wednesdays, and Fridays
Location: B235 Van Vleck Hall
Instructor
Lu Wang
Office: 809 Van Vleck Hall
Email: luwang@math.wisc.edu
Office Hours
Time: 11am-Noon, on Mondays and Wednesdays
Location: 809 Van Vleck Hall
Textbooks
[L] Introduction to smooth manifolds (2nd Edition), by John M. Lee, ISBN 13: 9780387954950(required)
[W] Foundations of differentiable manifolds and Lie groups (2nd Edition), by Frank W. Warner(required)
[S] A comprehensive introduction to differential geometry; Vol. 1 (3rd Edition), by Michael Spivak
Syllabus
Differentiable manifold, differential map
Tangent space, vector bundle
Immersion and embedding
Vector field, Lie derivative
Frobenius Theorem
Tensor and exterior algebra
Integration, Stokes' Theorem
De Rham cohomology, Poincare Lemma
Sard's Theorem, degree of a map
Homework
There will be homework assignments on a regular basis, mostly selected from the textbooks of Lee and Warner. Assignments are usually posted here about two weeks before the due date. Homework will be collected in beginning of the lecture on the due date and graded only on completeness. No late homework will be accepted.
HW#1 (due Sep. 22): [L] pp. 29-31: #1-1, 2, 4, 6, 7, 8, 9, 10
HW#2 (due Oct. 6): [L] pp. 48-19: #2-1, 5, 13, 14; pp. 75-76: #3-3, 4, 5, 6; pp. 95-97: #4-4, 5, 13
HW#3 (due Oct. 27): [L] pp. 123-124: #5-1, 4, 7, 9, 11, 17; pp. 147-149: #6-1, 6, 9, 11
HW#4 (due Nov. 10): [L] pp. 199-204: #8-6, 7, 9, 16; pp. 245-248: #9-1, 2, 4, 7
HW#5 (due Dec. 6): [L] pp. 245-248: #9-16, 18; pp. 324-326: #12-1, 3, 4, 5, 7, 8, 10; pp. 373-376: #14-1, 3, 4, 5, 6; pp. 397-399: #15-3, 5, 6; pp. 434-439: #16-2, 6, 9
HW#6 (not turn-in): pp. 434-439: #16-14, 18, 20, 22
Presentations
There will be three short, in-class presentations of solutions to homework assignments that have been collected. You will be assigned the problem(s) to present about a couple of days in advance. Make-up presentations will not be provided. Tentative presentation times are listed here.
Presentation I: Oct. 9 and 13, in class, based on HW#1 and 2
Presentation II: Nov. 13 and 15, in class, based on HW#3 and 4
Presentation III: Dec. 8 and 11, in class, based on HW#5
Grading
Homework: 10%
Presentation I: 30%
Presentation II: 30%
Presentation III: 30%
Letter grades will be assigned roughly according to the following standard:
A: 90+
AB: 86+
B: 77+
BC: 72+
C: 60+
D: 48+
F: 0+
Lecture Schedule
The following lecture schedule will be updated during the semester.
Week of Sep. 6
Introduction
Manifolds and smooth structures: [L] pp. 1-24
Week of Sep. 11
Smooth maps, diffeomorphisms: [L] pp. 32-40
Partition of unity: [L] pp. 40-47
Week of Sep. 18
Tangent vectors: [L] pp. 50-60
Coordinate representations: [L] pp. 60-65
Tangent bundles: [L] pp. 65-75
Week of Sep. 25
Maps of constant rank: [L] pp. 77-81
Rank theorem, embeddings: [L] pp. 81-88
Submersions, covering maps: [L] pp. 88-95
Week of Oct. 2
Embedded submanifolds: [L] pp. 98-103
Level sets, Immersed submanifolds: [L] pp. 104-112
Restricting maps to submanifolds: [L] pp. 112-114
Week of Oct. 9
Presentation I on Oct. 9 and 13
Week of Oct. 16
Tangent spaces to submanifolds: [L] pp. 115-120
Sard's theorem: [L] pp. 125-131
Week of Oct. 23
Whitney embedding theorem: [L] pp. 131-136
Tubular neighborhoods and transversality: [L] pp. 137-141, 143-147
Vector fields on manifold: [L] pp. 174-181
Week of Oct. 30
F-related vector fields: [L] pp. 181-185
Lie Brackets: [L] pp. 185-189
Integral curves and flows: [L] pp. 206-211
Week of Nov. 6
Fundamental theorem on flows: [L] pp. 211-215
Complete vector fields: [L] pp. 215-216
Lie derivatives: [L] pp. 227-230
Week of Nov. 13
Presentation II on Nov. 13 and 15
Tensor product of covector: [L] pp. 305-307
Abstract tensor product: [L] pp. 307-311
Week of Nov. 20
Covariant/Contravariant tensors: [L] pp. 311-313
Symmetric/Alternating tensors: [L] pp. 313-316
Tensor fields: [L] pp. 316-324
Pullbacks of tensor fields
Lie derivatives of tensor fields
Algebra of alternating tensors: [L] pp. 350-359
Differential forms, exterior derivatives: [L] pp. 359-366
No class on Nov. 24 due to thanksgiving recess
Week of Nov. 27
Orientation of vector spaces: [L] pp. 378-380
Orientation of manifolds: [L] pp. 380-388
Week of Dec. 4
Riemannian volume form: [L] pp. 388-392
Integration of differential form: [L] pp. 402-410
Stokes' theorem: [L] pp. 411-415
Presentation III on Dec. 8
Week of Dec. 11
Presentation III on Dec. 11