## Mathematics 444## Differential Geometry |

- extend many of the basic concepts and tools of multivariable calculus and linear algebra to the contexts of calculus on manifolds and Riemannian geometry.
- develop and demonstrate a level of expertise in mathematical reasoning appropriate to a challenging upper-level mathematics course.

- MIT OpenCourseWare hosts a rather similar course in differential geometry based on a highly regarded text by Manfredo do Carmo, 18.950 Differential Geometry, Spring 2005. Readings for that course include a handout on the inverse and implicit function theorems.
- John Lee's course materials for Math 310, Introduction to Mathematical Reasoning, at the University of Washington includes helpful advice on writing mathematics and on assessing the mathematical writing of others.
- Wikipedia entry for Jean Frédéric Frenet.
- Biography of Jean Frédéric Frenet.
- Biography of Elie Joseph Cartan.

- Math 444 MW Spring 2008 (pdf)

Much of our course will center on working through a substantial set of exercises taken from O'Neill's text. A computer algebra system may prove to be invaluable for the more computational exercises. Developing clear, clean, and elegant solutions to those problems, and explaining those solutions to each other, will take up much of our class time. You are then to write up your own versions of those solutions and hand them in as homework. The class discussions represent a community effort and may not extend to all of the exercises for which you are responsible. The homework for this course consists in writing a representation of your own understanding of the solutions to these exercises, whether discussed in class or not. Although the class sessions are collaborative and explorative, writing the homework solutions is an individual effort and is definitive. The goal is to engage and internalize the central mathematical issues of correctness, precision of expression, elegance, and style.

- Exercises for Math 430 TTh Spring 2006 (pdf)

- Chapter 1: Calculus on Euclidean Space
- Chapter 2: Frame Fields
- Chapter 3: Euclidean Geometry
- Chapter 4: Calculus on a Surface
- Chapter 5: Shape Operators
- Chapter 6: Geometry of Surfaces in R
^{n} - Chapter 7: Riemannian Geometry
- Chapter 8: Global Structure of Surfaces

Sewanee's tradition of cordial and constructive student-faculty relationships is one of its great strengths, and I am very happy to support that tradition. If you would like to talk with me, please make an appointment, either when you see me in class or in the hallways or in my office (WL120), or by email (cparrish@sewanee.edu), or by voice mail message (x1333).

Homework is due on Mondays and Wednesdays!

- 3 review examinations 60 %
- homework and such 20 %
- final examination 20 %