Instructor: Igor Zelenko
Office: Blocker 601J
e-mail: zelenko"at"math.tamu.edu
Home-page: /~zelenko
Class hours:
MATH439 T 2:20- 3:35 Blocker 624
Office hours: TR 12:45pm-2:00pm or by appointment (in Blocker 601J).
Text:. Manfredo P. Do Carmo, Differential Geometry of Curves
and Surfaces: Revised and updated Second Edition (Dover Books on
Mathematics (the furst edition of this book can be used as well);
Additional recommended text:
Lectures on Differential Geometry by Iskander A. Taimanov, EMS Series of Lectures in Mathematics, 2008.
Elementary Differential Geometry by Barret O'Neil, revised second
edition, ISBN-13: 978-0120887354, ISBN-10: 0120887355 (it will be used in class if we will arrive to Cartan's approach via differential forms)
Course Description. We will closely follow
Chapters 1-4 of Do Carmo's book (especially for examples and exercises)
and Part 1 of Taimanov's book (especially for theoretical material):
Local and global theory of parameterized curves; regular surfaces,
local coordinates, first fundamental form, orientation, area; Gauss
map, second fundamental form; Gauss Bonnet theorem. If the time will
permit elements of Riemannian Geometry (following
part 2 of Taimanov's book) and introduction to the Cartan method
of equivalence (following O'Neil book) will be discussed.
Prerequisite: MATH 308 and 323 (or equivalent) or approval of instructor
Necessary mathematical background: Good
knowledge of vector calculus (MATH221/251/253) and basic knowledge of
matrices (within what is taught
in
Differential Equations course MATH308)
Grading. Your grade will be determined by weekly home assignments
(40%) due
on Tuesdays of each week,
by one mid-term exams (30%) and cumulative final exam (30%). ALL EXAMS WILL BE TAKE-HOME EXAMS.
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
Dates of exams will be specified later.
The sections covered on a test will be announced well before the test.
Complaints about test grades must be made within two weeks of when the test
is returned to the class. Beyond that time, the grade will not be changed.
Tentative weekly schedule (below by DC x.x we denote the
corresponding section x.x in DoCarmo's book , by T x.x the
corresponding section x.x from Taimanov's book, and by ON x.x the corresponding section of O'Neil's book).
Week 1: Basic notions of theory of curves: regular
curves, tangent lines, arc length, parametriztion by arc length (DC
1.2-1,3, T 1.1) .Curvature of the curve, Frenet frame in R^2 (T 1.2);
Week 2: Frenet frame in R^3, torsion
(DC 1.5, T 1.3); The local canonical from of a curve up to a rigid
motion (DC 1.6); Frenet Frame in R^n (sketch of the construction);
Week 3 : Regular surfaces: implicit function
theorem, inverse function theorem , regular and critical values,
definiton of regular surface (DC 2.2,; T 1.4, T 2.1);
Week 4: Differntial functions on surfaces (DC 2.3); The tangent plane; the differential of the map (DC 2.4);
Week 5: The first fundamental form ; Area (DC
2.5, nd of T 2.1), Orientation of urvaces (DC 2.6). Gauss map (DC 3.2);
Week 6: Second fundamental form, principal curvature, Gaussian and mean curvatures (DC 3.2 continued, T 2.2-2.3);
Week 7: The Gauss map in local coordinates (DC 3.3);
Week 8: Isometries (DC 4.2); Gauss Egregium theorem, equations of compatibility (Gaus, Peterson-Mainardi-Codazzi), fundamental theorem of theory of sufraces (Bonnet theorem) (DC 4.3, T 2.1-2.5);
Week 9:Covariant derivative, patallel
transport; geodesics (DC 4.4, T 2.6). The Euler-Lagrange equation
(T 2.7);
Week 10: Gauss-Bonet theorem (DC 4.6, T 2.8) and its application such as Poincare index theorem (DC 4.6);
Week 11: Minimal surfaces (T 2.9, DC 3.5), Introduction to smooth manifolds (T 3.1-3.2, ON 4.8);
Week 12 (Nov 12, 14, 16): Tensors (T 3.3-3.4), Metric tensor (T 4.1); Differential forms (ON 1.6)
Week 13 (Nov 19): Affine connection and covariant derivative,
Chtistoffel symbols, torsion tensor (T 4.2)
Home assignments: Will be posted Tuesdays and is due next Tuesday.
The homework that is not submitted
by the end of lecture on Tuesday is "late" and will suffer a 20%
penalty if graded. You have until end of the lecture on next Tuesday to
deliver the late homework. The homework that is not submitted by
the end of lecture on next Tuesday will not be graded (you will get 0
points for it).
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.