# RIEMANNIAN GEOMETRY

### 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.

### Bibliography:

• Spivak, M., A Comprehensive Introduction to Differential Geometry, Vols. I-V, Publish or Perish, Berkeley (1979).

• Warner, F., Foundations of Differentiable Manifolds and Lie Groups, Springer, New York (1983).

### Week #1 Jan. 13-17

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

• Section 1.3, p. 7-8, Exercises: 1*, 2, 4, 5*, 6*, 7*
• Section 1.5, p. 12-13, Exercises: 9*, 10*, 13, 14
• Section 1.8, p. 20-21, Exercises: 16*, 17*, 18, 19*, 21, 23, 24
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:

• Section 2.4, p. 33-35, Exercises: 1*, 2*, 3, 4, 6, 7
• Section 2.6, p. 39-41, Exercises: 12a*, 13*, 14, 15*, 16, 17
• Section 2.9, p. 47-49, Exercises: 18, 19*, 20*, 21*, 22*, 23*, 24*, 25
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:

• Section 1.3 6
• Section 1.5 10
• Section 1.8 Show that (*u,*v) = (u,v) where u and v are smooth p-forms on an open set U of n-space. The inner product (u,v) is the one induced on p-forms, etc.
• Section 2.4 1, 6
• Section 2.6 13, 15
• Section 2.9 19, 21, 22
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.