Syllabus for MATH 622, Differential Geometry I,
  Spring 14

Instructor: Igor Zelenko
Office: Blocker 620B
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/S14M622.html

Class hours: TR 5:30-6:45 p.m. BLOC 160

Texts:

1. John M. Lee, Introduction to smooth manifolds, Edition: 1st Publisher:
Springer, Graduate Texts in Mathematics, Graduate Text in Mathematics, 218
ISBN: 0-387-95495-3 (hardcover) ISBN: 0-387-95448-1 (softcover)

2. Michael Spivak, A Comprehensive Introduction to Differential Geometry: Volume II, Chapter 6 and 7 there.

Prerequisite: Some basic concepts from Linear algebra and Topology. See, for example, the appendices of Lee's text.
The course does not have any graduate course as a prerequisite. The undergraduate prerequisites are standard Calculus courses (MATH 151, 152, 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 from the following chapters of Lee's text: 1 - 8, 11 - 14, 17-19. We will also go over the classical surface theory (both from a classical and a modern perspectives). 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 (corresponds to Spivak, volume 2, chapters 6 and 7).

Grading. Your grade will be determined by  biweekly home assignments (40%)  and two midterm exams and final exam  (20% 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 exams will take place in evenings,  out of regular class. The first midterm will be around the  end of February and the second one around the beginning of April.  The exact dates/time/location of them will be announce well in advance.

Tentative course schedule:

Jan 14-16-21  Lee Chap. 1-2, Topological and smooth  manifolds, smooth functions and maps, partition of unity, Lee groups (basic)

Jan  23-28       Lee Chap. 3-4,  Tangent Vectors, Pushforwards, Computations in Coordinates, Tangent Vectors to Curves. Tangent Bundle, Vector Fields on Manifolds, Lie Brackets, Lie Algebra of a Lie Group.

Jan  30-Feb 4  Lee Chap. 5-6,  Vector Bundles (briefly). Covectors, Tangent Covectors on Manifolds, Cotangent Bundle, Differential of a Function, Pullbacks, Line Integrals.

Feb 4-6-11        Lee Chap. 11-12,  Tensors, Differential forms, Exterior Differential Calculus.

Feb 13-18       Lee Chap. 7-8  Submersions, Immersions, Embeddings, Submanifolds.

Feb 20-25-27  Lee Chap 17-18-19 Integral curves and flow (briefly), Lie derivatives, Distributions, Involutivity, Frobenious Theorem.

March 4-6-18 (notes will be posted)  Extrinsic Geometry of surface  in R^3

March 20*  (notes will be posted) Generalization to hypersurfaces and submanifolds of R^n (brief)

March 25-27 (notes will be posted) Riemannian Geometry of surfaces.

April 1-3-8-10**( related to Spivak, v. 3, Chap 6-7)  Affine connections,  covariant derivatives, parallel transport, the torsion and curvature tensor, Levi-Civita connection, the torsion and curvature forms, Bianchi identity.

April  15-17 Lee Chap 13-14, Orientations, integration obn manifolds, Stokes theorem.

April 22-24 (notes will be posted) Gauss-Bonnet Theorem

* Might be ommitted if we will be short in time)

** Might be swapped with the subsequent material (which is important for the  qualifying exam)


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, in Cain Hall, Room B118, or call 845-1637. For additional information visit http://disability.tamu.edu.