Syllabus for MATH 623, Differential Geometry II,
  Spring 16

Instructor: Igor Zelenko
Office: Blocker 601J
Office hours (subject to change):  MW 2:15-3:45 p.m., F 10:00-11:10 a.m.  or by appointment.
e-mail: zelenko"at"math.tamu.edu
Home-page: /~zelenko
Home-page of the course: /~zelenko/F16M623.html

Class hours: MWF 12:40-01:30 p.m. BLOC 121

Texts:


1. Manfredo Perdigao do Carmo (Author) ( Francis Flaherty (Translator)), Riemannian Geometry, 1st Edition, Burkhauser Boston, 1992

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

 We will also discuss some additional topics and the material on the topics will be distributed in class.

Prerequisite: Some basic concepts from Linear algebra, Topology, and Analysis on manifolds.
The course Differential Geometry I, M622, is strongly recommended but not required. 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:

1. S. Kobayashi, K. Nomizu, Foundations of  Dofferential Geometry, Vol 1 (1963) and 2 (1969), Interscience publisher.

2. S. Sternberg,  Lectures on Differential Geometry, AMS Chelsey Publishing, 2nd Edition

3. (regarding the generalized Gauss-Bonnet Theorem)  Michael Spivak, A Comprehensive Introduction to Differential Geometry: Volume V , chpter 13.

 

Material. This is the second semester of a year-long graduate course in differential geometry. We will discuss the theory of affine connections, covariant derivatives, parallel transport, geodesics,torsion , Levi-Civita connection, curvature tensors and its properties, Bianchi identities (chapters 1-4 of Do Carmo together with the  Chapter 6 of Spivak, vol. 2), the Cartans's approach via moving frames  (Chapter 7 of Spivak , vol 2), connections in principle bundles ( Chapter 8 of Spivak), Jacobi fields and conjugae points ( chapter 5 of Do Carmo), spaces of constant curvature (Chapter 8 of Do Carmo), Comparison Theorems in Riemanian Geometry (chapters 9, 10, and possibly 11),. If the time will permit we will discuss some elements of Conformal Geometry, Kahler geometry, and generalized Gaus-Bonnet theorem.

Grading. Your grade will be determined by  home assignments given once in  2-3 weeks  (70%)  and a presentation (30%)  at the end of the semester that will be assigned to you in advanced on a topic which continues some topic discussed in a class or on a new topic which was not covered in class. 

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

Grade complaints: If you think a homework 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: There will not be exams in this class.
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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students with disabilities be guaranteed a learning environment that provides for reasonable accommodation of their disabilities. If you believe
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