Syllabus for MATH 622, Differential Geometry I,
Spring 16
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/S16M622.html
Class hours: TR 2:20-3:35
p.m. BLOC 121
Texts:
1. Frank W. Warner, Foundations of Differentiable Manifolds and Lie Groups , Springer Graduate Texts in
Mathematics, v. 94.
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.
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:
- Good reference for the Inverse and Implicit
Function Theorems and Exterior Differential Calculus: Michael Spivak , Calculus
on Manifolds.
- For undegraduate level Differential Geometry of curves and surfaces (which will be also covered in our course): Iskander A. Taimanov, Lectures on Differential Geometry by , EMS Series of Lectures in Mathematics, 2008 and Manfredo P. Do Carmo, Differential geometry of curves and
sufaces, Prentice Hall, 1976
- Michael Spivak , A
Comprehensive Introduction to Differential Geometry, v.1 (in addition to the material of Warners' book).
Material. This is the first semester of a year-long graduate
course in differential geometry. We will cover material from the
chapters 1,2, 4 of Warners's tex and the some part of chapter 3. We
will also
go over the classical curves and 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 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 20-25) . 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 Warner Chapter 1 Preliminaries, Differentiable manifolds, The second axion of countability and partition of unity
Week 2 Warner Chapter 1 Tangent
Vector and Differnetials. Submanifolds, Diffeomorphisms, and Inverse Function Theorem
Week 3 Warner Chapter 1 Implicit Function Theorem. Vector Fields
Week 4 Warner Chapter 1 Distributions and Frobenius Theorem
Week 5 Warner Chapter 2 Tensors and Exterior Algebra
Week 6 Warner Chapter 2 Tensor Fields and Differential Forms
Week 7 Warner Chapter 2 The Lie Derivative. Differential Ideals
Week 8 Warner Chapter 4 Orientation and Integration on manifolds
Week 9 Warner Chapter 4 Integration on manifolds continued, de Rham cohomology
Week 10 Warner Chapter 3 Lie Groups and Their Lie algebras, Homomorphisms, Lie subgroups
Week 11 Spivak, v. 2, Chapter 1 Starting from Frenet-Serres formula and Chapter 3 Part B Gauss's theory of surfaces
Week 12 Spivak, v. 2, Chapter 3 Part B Gauss's theory of surfaces (continued) and Gauss-Bonnet formula (see also Taimanov or Do Carmo for the latter)
Week 13 Spivak, v. 2, Chapter 6 The nabla operator (brief)
Week 14 Spivak, v. 2, Chapter 7 The moving frame (brief)
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].