Differential Geometry I (first semester)

Professor: Luis Fernandez	Office & Telephone: Rm. 4213, (212) 817-8561.
Days, times, and classroom: Tu 4:15 – 5:45pm, Th. 4:30 – 6pm, Rm 6421	Office hours: Tu, Th 2-3pm, or by appointment.
Course page: http://fsw01.bcc.cuny.edu/luis.fernandez01	e-mail: lfernandez1@gc.cuny.edu

Overview

In this course we will learn the “geo” part of “geometry”. In other words, we will not “measure” anything, except for some volume at the end. We will learn the definitions and language that will later permit the introduction of metrics, curvature, etc. in the second semester of the course.

We will follow the book by Loring Tu, “An Introduction to Manifolds”. It is concise and emphasizes the concepts that I want you to learn very well. My goal that you learn the basics very well so that you can then expand your knowledge easily on your own.

In addition, we will complement with John Lee’s “Introduction to Smooth Manifolds”. This book is much more thorough and a great source to consult and look for additional topics. But it is also harder to read. The sections, assigned exercises, etc are in the daily syllabus below for both textbooks.

It is imperative that you study each assigned chapter in Tu’s book, and do the exercises within the text as well as the assigned exercises. After that, go ahead and read the corresponding chapter in Lee’s book.

Do your best to prepare each class by reading ahead. Even if you do not go over all the details, it will be much easier later.

Do all the assigned exercises. I will not collect these, but you will have a written homework to hand-in every week.

The final grade will be based on the homework assignments as well as a final exam at the date and time we arrange at the end of the semester.

Books

In addition to the books above, these are other sources that you can check.

Foundations of Differentiable Manifolds and Lie Groups, by Frank W. Warner.
A Comprehensive Introduction to Differential Geometry, Vol 1, by Michael Spivak.

Daily Plan

	Chapters from L. Tu	Read from J. M. Lee	Exercises	Complements	Homework
08/29/24	Ch. 1. Smooth Functions on a Euclidean Space Ch. 2.Tangent Vectors in Rn as Derivations	Ch. 3, only “Geometric Tangent Vectors”	Tu: 1.1, 1.2, 1.5, 1.7, 2.1, 2.3, 2.4. Lee: 1-6, 1-7, 2-3.	Appendix A of L. Tu: Point-Set Topology
09/03/24	Ch. 3. Alternating k-Linear Functions	Ch. 12, up to “Symmetric and Alternating Tensors”	Tu: 3.1, 3.3, 3.4, 3.5, 3.7.
09/05/24	Ch. 3. Alternating k-Linear Functions	Ch. 12, up to “Symmetric and Alternating Tensors”	Tu: 3.9, 3.10, 3.11. Lee: 3-1, 3-8, 14-1, 14-2, 14-3	General definition of tensor product (Lee, p. 307 ff)
09/10/24	Ch. 4. Differential Forms on Rn	Ch. 14, only “The Algebra of Alternating Tensors”	Tu: 4.1 to 4.5.
09/12/24	Ch. 4. Differential Forms on Rn	Ch. 14, only “The Algebra of Alternating Tensors”	Tu: 4.6 to 4.9. Lee: 14-6.		Hw 1 due.
09/17/24	Ch. 5. Manifolds Ch. 6. Smooth Maps of a Manifold	Ch. 1. Smooth Manifolds, except “Manifolds with Boundary” Ch. 2. Smooth Maps, except “Partitions of Unity” (seen later).	Tu: 5.1 to 5.4. 6.1 to 6.8
09/19/24	Ch. 7. Quotients		Tu: 7.1, 7.2, 7.3	Read The Quotient Manifold theorem (Th. 21.10 in Lee) and all the definitions needed to understand it. You can use this theorem in the future to prove that a space is a manifold.	Hw 2 due.
09/24/24	Ch. 8. The Tangent Space	Ch. 3. Tangent Vectors, except “The Tangent Bundle” (seen later).	Tu: 8.1 to 8.5
09/26/24	Ch. 8. The Tangent Space	Ch. 3. Tangent Vectors, except “The Tangent Bundle” (seen later).	Tu: 8.6 to 8.9		Hw 3 due.
10/01/24	Ch. 9. Submanifolds	Ch. 5. Submanifolds	Tu: 9.1 to 9.3. 9.4 and 9.5 (find adapted charts), 9.6 to 9.9
10/03/24	No class.
10/08/24	Ch. 10. Categories and Functors		Tu: 10.1 to 10.5		Hw 4 due.
10/10/24	Appendix B. The Inverse Function Theorem on Rn and Related Results Ch. 11. The Rank of a Smooth Map	Ch. 4. Submersions, Immersions, Embeddings	Tu: Appendix B, all, 11.1 to 11.5.
10/15/24	No class. Monday schedule
10/17/24	Ch. 12. The Tangent Bundle	Ch. 3. “The Tangent Bundle” Ch. 10. Vector Bundles	Tu: 12.1, 12.2. Lee: 5-1		Hw 5 due.
10/22/24	Ch. 12.The Tangent Bundle	Ch. 3. “The Tangent Bundle” Ch. 10. Vector Bundles	Lee: 10-5, 10-6, 10-7, 10-10	More about vector bundles
10/24/24	Ch. 13. Bump Functions and Partitions of Unity	Ch. 2, “Partitions of Unity”	Tu: 13.1 to 13.6. Lee: 2-14		Hw 6 due.
10/29/24	Ch. 14.Vector Fields	Ch. 8. Vector Fields, except “The Lie Algebra of a Lie Group” (seen later)	Tu: 14.1 to 14.7 Lee: 8-6, 8-7, 8-10, 8-11, 8-13, 8-16, 8-19, 8-20,
10/31/24	Ch. 14.Vector Fields	Ch. 9. Integral Curves and Flows, up to “Commuting Vector Fields” (seen later)	Tu: 14.8 to 14.13. Lee: 9-3, 9-4, 9-17, 9-18, 9-19		Hw 7 due.
11/05/24	Ch. 15. Lie Groups	Ch. 7. Lie Groups	Tu: 15.1 to 15.15 Lee: 7-2, 7-4, 7-11, 7-16, 7-17.
11/07/24	Ch. 16. Lie Algebras	Ch. 8. “The Lie Algebra of a Lie Group”	Tu: 16.1 to 16.10 Lee: 8-24, 8-28, 8-29, 8-30.	Exp in general Lie Group and theorems. Lee, Ch 20	Hw 8 due.
11/12/24	Ch. 17. Differential 1-Forms	Ch. 14.Differential Forms, only “Differential Forms of Manifolds”	Tu: 17.1 to 17.3
11/14/24	Ch. 18. Differential k-Forms	Ch. 14.Differential Forms, only “Differential Forms of Manifolds”	Tu: 18.1 to 18.8. Lee: 14-1, 14-2, 14-3, 14-4, 14-5.		Hw 9 due.
11/19/24	Ch. 19. The Exterior Derivative	Ch. 14, “Exterior Derivatives”	Tu: 19.1 to 19.6. Lee: 14-1, 14-2, 14-3, 14-4.
11/21/24	Ch. 19. The Exterior Derivative	Ch. 14. “Exterior Derivatives”	Tu: 19.7 to 19.12 Lee: 14-5, 14-6, 14-7.		Hw 10 due.
11/26/24	Ch. 20. Orientations	Ch. 15. Orientations.	Tu: 20.1 to 20.7. Lee: 15-1, 15-2, 15-3, 15-5 (assume without boundary),15-7.
11/28/24	No class. Thanksgiving
12/03/24	Ch. 21. Manifolds with Boundary	Ch. 1, “Manifolds with Boundary”	Tu: 21.1 to 21.6. Lee: 1-11, 1-12.
12/05/24	Ch. 22. Integration on a Manifold	Ch. 16. Integration on Manifolds.	Tu: 22.1 to 22.3. Lee: 16-1, 16-2, 16-4, 16-6, 16-10,		Hw 11 due.
12/10/24	Ch. 23. De Rham Cohomology	Ch. 17. De Rham Cohomology.	Tu: 23.1. Lee: 17-1, 17-4, 17-10.
12/12/24	Sard? Frobenius?				Hw 12 due.
?	Final Exam