MAT367: Differential Geometry

General Info

• Week #1

Tuesday, May 4
Lecture notes	In-class notes
Summary	Introduction Finding the best definition of a surface in ℝ3. Informal discussion of an abstract manifold and why we need a definition that doesn't see an ambient space. Informal introduction to differential topology and differential geometry.
Sections covered	---
Readings	Recommended before next lecture: Appendix A.1-A.8
Tutorial	---

Thursday, May 6
Lecture notes	In-class notes
Summary	Definition of k-dim manifolds in ℝn. Elements of Point set topology. Definition of a topological manifold and a smooth manifold.
Sections covered	Parts of Appendix A. sections 5.1-5.3
Readings	Mandatory: All of Appendix A

• Week #2

Tuesday, May 11
Lecture notes	In-class notes
Summary	Review of the definition of a smooth manifold Definition of a smooth maximal atlas Some examples of smooth manifolds: Euclidean space, k-dim manifolds in ℝn
Sections covered	5.3
Readings	Recommended before next lecture: 7.1-7.3 (pg 71-73)
Tutorial	Apendix A

Thursday, May 13
Lecture notes	In-class notes
Summary	More examples of manifolds: level sets, S 1, Product manifolds Informal discussion of the gluing procedure Quotient spaces and criterion for Hausdorff
Sections covered	Section 7.1-7.5
Readings	---

• Week #3

Tuesday, May 18
Lecture notes	In-class notes
Summary	Criterion for Hausdorff and second countable Projective space Definition of a smooth function on a manifold
Sections covered	7.6, 6.1
Readings	---
Tutorial	Standard atlas on the projective space (section 7.7)

Thursday, May 20
Lecture notes	In-class notes
Summary	Definition of smooth maps from manifolds to manifolds Examples of different smooth structures on ℝ Inverse function theorem on manifolds Intuition of the derivative of a map from ℝ n to ℝ m Definition of the tangent space
Sections covered	6.2-6.4, 6.6-6.7, 8.1 (partially)
Readings	Optional: 6.5

• Week #4

Tuesday, May 25
Lecture notes	In-class notes
Summary	The tangent space on manifolds Definition of the differential of a smooth map
Sections covered	2.1-2.3, 8.1-8.2
Readings	Optional: Section 2.1-2.3
Tutorial	6.5, end of 6.7, and some probems from section 6.

Thursday, May 27
Lecture notes	In-class notes
Summary	Chain rule and its corollaries Dimension of a manifold = dimension of the tangent space The coordinate basis for the tangent space The differential in coordinates Curves on manifolds
Sections covered	8.2-8.7
Readings	---

• Week #5

Tuesday, June 1
Lecture notes	In-class notes
Summary	Immersions, submersions, and embeddings Constant rank theorem Embedded submanifolds and examples Whitney's embedding theorem
Sections covered	8.8, 11.1-11.2
Readings	---
Tutorial	Problems 8.1, 8.2, 8.7, 11.1, 11.2., 11.3 Tangent space of a product manifold

Thursday, June 3
Lecture notes	In-class notes
Summary	Regular submanifolds The regular and constant level set theorem Tangent space of a submanifold The tangent bundle and its smooth structure Smooth sections of TM and smooth vector fields
Sections covered	9.1-9.3, 12.1, 12.2, 12.4
Readings	Mandatory:9.4 Optional: 11.4, 11.5 Recommended before next lecture: Section 13

• Week #6

Tuesday, June 8
Lecture notes	In-class notes
Summary	Existence of bump functions (weak and strong version) Existence of partition of unity Smooth Extension lemma (weak and strong version) Extension of a coordinate vector field
Sections covered	Section 13
Readings	Optional: section 13 and appendix C
Tutorial	Application of the regular level set theorem Problem 12.2

Thursday, June 10
Lecture notes	In-class notes
Summary	Review of the tangent bundle and smooth sections Equivalence of the two definitions of vector fields, smooth sections and derivations Two smoothness criterion for vector fields Integral curves of a vector field
Sections covered	14.1-14.2, 2.4-2.5
Readings	Optional: 12.5

• Week #7

Tuesday, July 6
Lecture notes	In-class notes
Summary	Review of integral curves The local flow of a vector field Existence of a maximal flow of a vector field on a coordinate set Fundamental theorem of flows
Sections covered	14.2-14.3
Readings	---
Tutorial	Computing flows. Computation exercises involving the Lie bracket.

Thursday, July 8
Lecture notes	In-class notes
Summary	Lie derivative Lie bracket and Lie algebras
Sections covered	14.4, 20.2
Readings	Mandatory: 14.5, 14.6

• Week #8

Tuesday, July 13
Lecture notes	In-class notes
Summary	The statement of Frobenius theorem
Sections covered	---
Readings	Optional: Lecture notes from past 367 courses here , section 6.6 Section 4.8 in "An introduction to differentiable manifolds and Riemannian geometry" by William Boothby
Tutorial	Geometric interpretation of the Lie bracket (and some compuational questions) Examples of an integrable and a non-integrable distribution

Thursday, July 15
Lecture notes	In-class notes (corrected)
Summary	Proof of Frobenius theorem. The contangent bundle Definition of smooth 1-forms
Sections covered	17.1, 17.3
Readings	Optional: Review differential forms on Rn. (Sections 3 and 4)

• Week #9

Tuesday, July 20
Lecture notes	In-class notes
Summary	Smoothness criterion lemma for 1-forms Equivalent definition of a 1-form
Sections covered	17.2, 17.4
Readings	Recommended before next lecture: Sections 3 and 4 (differential forms on Rn)
Tutorial	Pull back of 1-forms and its propeties Computation Examples Section 19.7: Constructing a nonvanishing 1-form on the circle

Thursday, July 22
Lecture notes	In-class notes (corrected)
Summary	The pull back of 1-forms and its properties The algebra of alternating k-tensors The bundle of alternating k-tensors Differential k-forms
Sections covered	17.5, 18.1-18.4, 18.6
Readings	---

• Week #10

Tuesday, July 27
Lecture notes	In-class notes
Summary	Pull-back of a k-form Motivation for the exterior derivative
Sections covered	Section 18.5
Readings	Optional: 18.7
Tutorial	Section 17.6 and Problem 17.1 Problem involving the action of a 2-form on vector fields

Thursday, July 29
Lecture notes	In-class notes
Summary	Existence and uniqueness of the exterior derivative Global intrensic formula for the exterior derivative
Sections covered	section 19.1-19.5, 20.6
Readings	Optional: 19.6-19.7

• Week #11

Tuesday, August 3
Lecture notes	In-class notes
Summary	Interior multiplication Lie derivative on differential forms
Sections covered	20.1-20.4
Readings	---
Tutorial	Computation examples for interior multiplication, Lie derivative, and exterior derivative.

Thursday, August 5
Lecture notes	In-class notes
Summary	The properties of the lie derivative on differential forms Cartan calculus Orientation on vector spaces
Sections covered	20.5-20.6, 21.1-21.2
Readings	Recommended before next lecture: sections 21 and 22

• Week #12

Tuesday, August 10
Lecture notes	In-class notes
Summary	Orientation on manifolds Manifolds with boundary
Sections covered	21.3-21.5, 22.1-22.2
Readings	Recommended before next lecture: Section 22
Tutorial	Orientation perserving (computation question) Boundary orientation

Thursday, August 12
Lecture notes	In-class notes
Summary	Vector fields and differential forms on manifolds with boundary The boundary orientation for manifolds with boundary The role of alternating tensors in the theory of integration Integration on theorem Stokes theorem
Sections covered	22.3- 22.6, 23.1-23.5
Readings	Optional: 23.6

1. Assignment 1 (Due on Friday, May 14 by 11:59 pm)
   Typos fixed (May 9, 1:10 PM)
   
2. Assignment 2 (Due on Sunday, May 30 by 11:59 pm)
3. Assignment 3 (Due on Friday, June 11 by 11:59 pm)
   Edited (June 6, 11:00 PM)
   
4. Assignment 4 (Due on Sunday, July 11 by 11:59 pm)
   Typos fixed (July 14 12:56 PM)
   
5. Assignment 5 (Due on Friday, July 23 by 11:59 PM)
   Typos fixed (July 23, 11:00 AM, problem 2h)
   
6. Assignment 6 (Due on Friday, August 6 by 11:59 PM)
   
7. Assignment 7 (Due on Friday, August 13 by 11:59 PM)
   Typos fixed (August 13, 3:11 PM)
   

Mock Midterm

Midterm

Mock Exam

Exam