Syllabus for MATH 623, Differential Geometry II,
  Fall 19

Instructor: Igor Zelenko
Office: Blocker 601J
Office hours (subject to change): Monday and Friday 11am-noon, Wednesday 3-4pm or by appointment in my office
e-mail: zelenko"at"math.tamu.edu
Home-page: https://www.math.tamu.edu/~zelenko/
Home-page of the course:

Class hours: MWF 1:50-2:40pm. BLOC 205B

Main Text:

Liviu  Nicolaescu, Lectures on Geometry of Manifolds, 2nd Edition, World Scientific Publishing, 2007


Other texts:

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

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

3.  2. R.W. Sharpe, Differential Geometry:Cartan's Generalization of Klein's Erlangen Program,  Springer Graduate Texts in Mathematics, Vol. 166.

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

The material that not included on the texts above will be posted in the form of the lecture notes.

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. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, any edition (there are 3 editions)


 

Material. This is the second semester of a year-long graduate course in differential geometry. We will discuss the theory connections on vector bunbdles :covariant derivatives, connection forms, parallel transport,curvature tensor, torsion , Bianchi identities  (Nicolaescu , subsection 3.3), elements of Riemannian Geometry :Levi-Civita connection, goedesics, properties of Riemannian curvature tensor, elements of Cartan's method of moving frames and Geometry of submanifolds of Riemannian manifolds, the Gauss-Bonnet theporem for oriented surfaces (Chapter 4 of Nicolaescu), The variational theory of Geodesics: Jacobi fields, conjugate points, Bonnet-Myers and Racuh comparison theorems (Section 5,2 of Nicolaescu and Chapters 9 and 10 of DoCarmo),  Elements of Cohomology theory:  De Rhan cohomology, the Poincare duality and elements of intersection theory ( Nicolaescu, subsections  7.1, 7.2, and 7.3), Characteristic classes: Connection in principle bunde, G-structures , Invariant polynomials, , the  Chern-Weil theory, Chern classes, Pontryagim classes, Euler class, Gauss-Bonnet-Chern theorem (Nicolaescu, Chapter 8, for connectionss in principle bundles  and elements of G-structures  the additional sources are Chapter 8 of Spivak and Chapter VII of Sternberg). If the time will perit we wil discuss prolongation f G-structures, Cartan connection with applications to Conformal and Projective Geometry (the correspondong sources and materials will be provided).

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. You also can suggest a topic related to your own research and Differential Geometry.

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.


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].


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