Math 622, Spring 2017Differential Geometry I

Instructor: Dean BaskinOffice: Blocker 614BE-mail: dbaskin@tamu.edu

Lectures: MWF 9:10–10:00 in Blocker 161Office Hours: MW 1:30–2:30pm

and by appointment

Exams.

• Midterm exam: Wednesday, March 22, 2017, in class.• Final exam: Friday, May 5, 2017, 8–10am.

Course description and prerequisites.

Description. This is the first semester of a year-long graduate course in differential geometry. Thetopics covered include: curves and surfaces in R3 and generalizations to submanifolds of Euclideanspace; smooth manifolds and mappings; differential forms; Lie groups and algebras; Stokes’ theorem;de Rham cohomology; the Frobenius theorem; Riemannian manifolds.

Prerequisites. This course does not have any graduate course as a prerequisite. The only officialprerequisites are MATH 304 (linear algebra) and the approval of the instructor. We will use freelysome basic concepts from linear algebra and topology as well as those from the standard calculuscourses and differential equations. I will give all mathematical background beyond the above coursesin class.

Textbook.

Required texts.

(1) Michael Spivak, A Comprehensive Introduction to Differential Geometry: Volume I. We willcover most of this book.

(2) Michael Spivak, A Comprehensive Introduction to Differential Geometry: Volume III. Wewill cover chapter 2 and some other sections of this book.

We will also use some supplementary notes for the analysis of curves in R3. These will be providedat the appropriate time.

Optional texts. Additional references you may like to consult include:

• Reference for inverse and implicit function theorems and exterior differential calculus:– Michael Spivak, Calculus on Manifolds.

• Undergraduate level differential geometry of curves and surfaces (also covered in our course):– Manfredo P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall,

1976.– Iskander A. Taimanov, Lectures on Differential Geometry, EMS Series of Lectures in

Mathematics, 2008.• Other perspectives on the material in the required texts:

– Frank W. Warner, Foundations of Differentiable Manifolds and Lie Groups, SpringerGraduate Texts in Mathematics, v.94.

– John M. Lee, Introduction to Smooth Manfiolds, Springer Graduate Texts in Mathemat-ics, v.218.

Course website. The course website is http://www.math.tamu.edu/~dbaskin/math622-spring17/.1

2

Grading policies. Your grade will be determined by homework assignments (40%), one midtermexam (30%), and the final exam (30%). The grading will be at least as generous as the following scale:

• A: 85%–100%• B: 75%–85%• C: 65%–75%• D: 55%–65%

Tentative course schedule. Note that the last two weeks of the course have nothing scheduled forthem. This provides some slack in the schedule in case we spend more time on some subjects. If westick to the schedule, then we will have a (brief) introduction to Riemannian geometry in those weeks.

• Week 1: Preliminaries and differentiable manifolds. (Chapters 1, 2 of volume I)• Week 2: Diffeomorphisms and the inverse and implicit function theorems (Chapter 2)• Week 3: Tangent vectors, the tangent and cotangent spaces. (Chapters 3, 4)• Week 4: Tensors (Chapter 4)• Week 5: Vector fields (Chapter 5)• Week 6: Distributions and the Frobenius theorem (Chapter 6)• Week 7: Differential forms (Chapter 7)• Week 8: Integration on manifolds (Chapter 8)• Week 9: de Rham cohomology (Chapter 8)• Week 10: Lie groups (Chapter 10)• Week 11: Curves and surfaces (Chapter 2 of Volume 3 as well as provided notes)• Week 12: Surfaces (Chapter 2 of Volume 3)

Course policies. If you believe that a homework or exam has been graded incorrectly, you have oneweek from the time the graded assignment was returned to you to bring the issue to the instructorsattention.

Americans with Disabilities Act (ADA) policy statement. The Americans with Disabilities Act (ADA)is a federal anti-discrimination statute that provides comprehensive civil rights protection for personswith disabilities. Among other things, this legislation requires that all students with disabilities beguaranteed a learning environment that provides for a reasonable accommodation of their disabilities.If you believe you have a disability requiring an accommodation, please contact the Disability ServicesOffice, Department of Student Life, in Room B118 of Cain Hall.

Academic integrity statement and policy. Scholastic dishonesty will not be tolerated. Aggie HonorCode: An Aggie does not lie, cheat, or steal or tolerate those who do. For additional information, visithttp://aggiehonor.tamu.edu.

Copyright. Course materials should be assumed to be copyrighted by the instructor who wrote themor by the University.