Syllabus for MATH 622, Differential Geometry I,
  Spring 18

Instructor: Igor Zelenko
Office: Blocker 601J
Office hours (subject to change):  TR 10.00-11:30 a.m.  or by appointment.
e-mail: zelenko"at"math.tamu.edu
Home-page: /~zelenko
Home-page of the course: /~zelenko/S18M622.html

Class hours: TR 2:20-3:35 p.m. BLOC 148


I will not follow any specific book but I will use several books and B. Dubrovin lecture notes (the link to the
latter is also below). The electronic versions of all this will be posted on
eCampus, so you are not required to purchase it.

1. John M. Lee, Introduction to smooth manifolds, Publisher:
Springer, Graduate Texts in Mathematics, Graduate Text in Mathematics, 2nd edition ISBN-13: 978-1441999818
ISBN-10: 1441999817 (you also can use the 1st edition)

2. Michael Spivak, A Comprehensive Introduction to Differential Geometry: Volume I and Volume II.

3. Foundations of Differentiable Manifolds and Lie Groups (Graduate Texts
in Mathematics) (v. 94)  by Frank W. Warner.

4. the link to B. Dubrovin lecture notes are

http://people.sissa.it/~dubrovin/dg.pdf

Additional references that you may like to consult include (will be posted as well):

 Manfredo P. do Carmo, Differential Forms and Applications (good reference  for Exterior differential Calculus, basics
of the method of moving frames and Gauss-Bonnet 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.


Additional references that you may like to consult include:

Material. This is the first semester of a year-long graduate course in differential geometry.  We will cover material on basics of manifolds, partition of unit, smooth maps and thei differentials, vector fields, Lie brackets, tangent and cotangent bundle, submanifiolds,  tensors and tensor fields, exterior differential, distributuons and theor yintegrability (Frobenious theorem), Lie derivatives, integration on manifolds and generalized Stockes' theorem,  elements of Lie groups and Lie algebras.  We will also go over the classical curves and surface theory , both from a classical and a modern perspectives, including Gauss-Bonnet Theorem. Hopefully we will also cover  some theory of affine connections, including covariant derivatives, parallel transport, the torsion and curvature tensor, Levi-Civita connection etc and the same from the point of view of the method of moving frames.

Grading. Your grade will be determined by  biweekly home 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 an evening,  out of regular class. The  midterm will be in the first week after the Spring break (the week of March 19-23) . The exact dates/time/location of the midterm will be announce well in advance.

Tentative course schedule (here a week means two successive  odd/even classes in case of cancellation of classes following by make-up classes, we will have 14 weeks=28 classes)

Week 1 Review of differential maps from R^n to R^m, the notion of  their differential and Jacobi matrix, Implicit and Inverse Function Theorems, Differentiable manifolds, Smooth maos between manifolds.

Week 2  Tangent Vector and Differentials, Vector fields, Lie brackets of vector fields;

Week 3  Embedded and Inverse submanifolds. Implicit and function theorems  revisited;

Week 4  Distributions and Frobenius Theorem

Week 5 Tensors and Exterior Algebra

Week 6 Tensor Fields and Differential Forms

Week 7 The Lie Derivative. Differential Ideals, Frobenious theorem in terms of differential forms.

Week 8 Orientation and Integration on manifolds

Week 9  Integration on manifolds continued, de Rham cohomology

Week 10 Lie Groups and Their Lie algebras, Homomorphisms, Lie subgroups

Week 11 Frenet frames for curves in Euclidean space  and Gauss's theory of surfaces

Week 12 Gauss's theory of surfaces (continued) and Gauss-Bonnet theorem;

Week 13  The theory of covariant derivatives and affine connections, Levi-Civita connections,  torsion and curvature tensors;

Week 14  Basic of Cartan methods of moving frames, torsion and curvature forms.
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.


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 persons with disabilities. Among other things, this legislation requires that all
students with disabilities be guaranteed a learning environment that provides for reasonable accommodation of their disabilities. If you believe
you have a disability requiring an accommodation, please contact Disability Services, currently located in the Disability Services building at the
Student Services at White Creek complex on west campus or call 979-845-1637. For additional information, visit http://disability.tamu.edu
[disability.tamu.edu].