RIEMANNIAN GEOMETRY

Math 623-600

Spring 1997




Instructor: Peter F. Stiller
Office: Milner Hall 215 or Blocker Bldg. 634B
Office Phone: 5-5727 or 2-2905
E-mail: stiller@math.tamu.edu
Office Hours: TBA



Course Information

Time: MWF 12:40 to 1:30
Place: Blocker 135
Text: Differential Forms and Connections by R. W. R. Darling, Cambridge University Press (1994).
Supplementary Material: See the bibliography below
Prerequisites: The equivalent of Math 622. The student should be familiar with concepts from linear algebra and multivariable calculus.
Grades: Mid-term Exam (25%), Final Exam (25%), Research Project (20%), Problem Sets (30%). The Mid-term Exam will be a take-home exam, while the Final Exam will be given in class. Homework problems will be assigned on a regular basis. This will include six special Problem Sets which will be turned in for a grade. The Research Project will involve outside reading and an oral report (below) .
Overview: This course will explore the geometry of differentiable manifolds. We will study a number of important mathematical objects, including vector bundles, Riemannian metrics, Riemannian manifolds, and connections.



Syllabus

Detailed Description: We will cover most of the text. Chapters 1-3 will be covered quickly, and Chapter 4, which is a review of Surface Theory (Math 622) will be omitted. The heart of the course consists of Chapters 5-7, which introduce many of the fundamental objects of study. Chapter 8 deals with integration on manifolds and the all important Stokes' Theorem. Finally, Chapter 9 introduces another important notion, i.e. connections. If time permits, we will look at some of the applications to physics in Chapter 10.

Topics



Bibliography:




Week #1 Jan. 13-17

Read Chapter 1, pgs. 1-23. Try the following exercises:

Note: Exercises marked with an asterisk are especially recommended.


Week #2 Jan. 20-24

Continue reading Chapter 1, pgs. 1-23. Also read the handout on Multilinear Algebra.


Week #3 Jan. 27-31

Read Chapter 2, pgs. 24-52. Try the following exercises:

Note: Exercises marked with an asterisk are especially recommended.

Section 2.11 on Maxwell's Equations will be particularly interesting to those with a background in physics. (This section is optional and may be omitted.)


Week #4 Feb. 3-7

Continue reading Chapter 2, pgs. 24-52.


Assignment #1

Do the following exercises for Chapters 1 and 2:

These problems will be due in class on Monday Feb. 17. (Note: 15 and 22 are definitely the hardest problems.)




Project Information

The major project consists of reading a paper (or other source) for information on one of the topics below and writing a brief synopsis of the key result(s). A 20 minute oral presentation will be given during the last week of the course.



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