Syllabus for MATH 622, Differential Geometry I,
  Spring 19

Instructor: Igor Zelenko
Office: Blocker 601J
Office hours (subject to change):  MW 10:30 a.m.-noon  or by appointment.
e-mail: zelenko"at"math.tamu.edu
Home-page: /~zelenko
Home-page of the course: /~zelenko/S19M622.html

Class hours: MW 12:45-2:00 p.m. BLOC 160

Main Texts:

Liviu  Nicolaescu, Lectures on Geometry of Manifolds, 2nd Edition, World Scientific Publishing, 2007

Supplementary Texts:

1. Frank Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer Graduate Texts in Mathematics, v.94

This book is very concise and contains in particular a very thorough treatment of the most material of the first three chapters of the book (and more) but it is too dry and abstract and will be used as the source for additional exercises and for references.

2. R.W. Sharpe, Differential Geometry:Cartan's Generalization of Klein's Erlangen Program,  Springer Graduate Texts in Mathematics, Vol. 166.

This text will be used for treatment of Frobenius theorem (chapter 2 there) and of Maurer -Cartan form on the Lie group  (Chapter 3 there) as the basis for Cartan's method of moving frame ( paragraph 4.2.3 of the main text).

3. M. Do Carmo, Differential Forms and Applications (Universitext).

This is an elementary and self-contained treatment of differential forms, manifold, generalized  Stokes theorem, and applications to intrinsic and extrinsic geometry of surfaces. It might be useful in addition to the main text if you feel that the treatment of the same subjects in the main text is too abstract.

4. John Lee, Introduction to Smooth Manifolds , Graduate Texts in Mathematics, Vol. 218),2nd Edition

This text will be used as the source  for  additional exercises and  for references. It is a very detailed and more elementary treatment of most of the first three chapters of the main text (except the section 3.3). It is recommended for those who like to see all details.

5. Michael Spivak , A Comprehensive Introduction to Differential Geometry, vol.1

This is the first volume of the famous classical 5 volumes series and it seems that the book of Lee was written based on this classical book. It will be used for references.You may enjoy reading this classics.

6. Michael Spivak , A Comprehensive Introduction to Differential Geometry, vol.2

This is the second volume of the famous classical 5 volumes series. It roughly contains the material of Section 3.3 and Chapter 4 of the main book.  The most valuable feature that the theory is given according to its historical development , from Gauss, Riemann, and Ricci to Cartan and Kozhul (including the translation of the original Gauss's and Riemann's works) . It will be used for references.You may enjoy reading this classics.

7. Walter Rudin, Principles of Mathematical Analysis, chapters 9 and 10 there

This text provides the analytic background for the course. In particular, chapter 9 contains all waht you need to know about differentiable function, contraction principles and proofs of  Inverse and Implicit function Theorems (that are only sketched in the main text). Chapter 10 contains the material on integration of differential forms and generalized Stockes' theorem.

 Prerequisite: Some basic concepts from Linear algebra and Topology.
The course does not have any graduate course as a prerequisite. The undergraduate prerequisites are standard Calculus courses (MATH 171/151, 172/152, 221/251 or equivalent), Differential Equations (MATH 308), and Linear algebra (MATH 304 or 323). I will give all mathematical background beyond the above courses in the class.

Material. This is the first semester of a year-long graduate course in differential geometry.  Our plan is to cover most of the first four  chapters of the Nicolaescu book. It is recommended to read the material before each class to be better  preapared to the lecture, as the lectures might be quite dense and fast (although I will try to adjust as much as possible the pace to your needs). I will assign what to read before each class.

Grading. Your grade will be determined by  weekly homework  assignments (40%)  and one midterm exams and final exam  (30% each).  The grade will be given according to the following percentage:

85%-100%=A, 75%-84%=B , 65%-74%=C, 55%-64%=D, less than 55%=F

Grade complaints: If you think a homework or exam was graded incorrectly you have one week from the time the graded assignment was returned to you to bring the issue to the instructor's attention. No complaints after that time will be considered.


Exams: The midterm exam will take place in some evening,  out of regular class. The  midterm will be in the first week after the Spring break (the week of March 18-22) . The exact dates/time/location of the midterm will be announce well in advance. The final exam is scheduled for Friday, May 3, 2019 10:30am-12:30pm in the regular classroom.

Homework: The Homework assignments will be posted usually before  Wednesday's evenings  with  submission on  next Wednesday in class.  Each assignment  will consist of 3-4 problems either mainly from Nocolaescu or my own but also from Warner or Lee.

E-campus: All announcements, course materials, and grades will be posted in eCampus and/or via email.


 Tentative course schedule (here a week means two successive  odd/even classes , we will have 14 weeks=28 classes)

Weeks 1-2
Some terminology and material from Warner  (Chapter 1, sections 2 and 5) and Lee (Chapters 4-5) will be also used here.

 Weeks 3-4-5  

Weeks 6-7-8  

   
Week 9

Nicolaescu,   Section 3.4 (except the last 2 sections there)  It is basically Integration of differential forms and generalized Stokes' theorem  (the material also is contained in Do Carmo Chapter 4, Lee, Chapter 16, Spivak, vol. 1 , Chapter 8, Rudin, Chapter 10).
 

Weeks 10-11

Nicolaescu, Section 3.3 Connections on Vector bundles.

Week 12-13-14
Nicolaescu, Chapter 4, Elements of Riemannian geometry

For subsections 4.2.3 (Cartan's moving frame) we also will use Sharpe Chapter 3, Section 1 on Maurer-Cartan form on Lie groups and also Sections 3 and 5 there.

Also for material of weeks 10-14 we recommend Spivak, vol. 2 Chapters 6 and 7.

In any case all what we will not cover on Riemannian geometry from chapter 4 of the main text will be covered and in more details in the second semester of the course.



Policy regarding absences related to injury or illness:

All such absences will be excused if sufficient documentation is provided as per University policy and
the instructor will help the student make up any missed material.

Scholastic dishonesty: Copying work done by others, either in-class or out of class, is an act of scholastic dishonesty and will be prosecuted to the full extent allowed by University policy. Collaboration on assignments, either in-class or out-of-class, is forbidden unless permission to do so is granted by your instructor. For more information on university policies regarding scholastic dishonesty, see University Student Rules .

Copyright policy: All printed materials disseminated in class or on the web are protected by Copyright laws. One xerox copy (or download from the web) is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited.


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