Syllabus for MATH 625 Section 1

Differentiable Manifolds I

Fall Semester 2016 CRN: 79292

Textbook: Manifolds and Differential Geometry,
 by Jeffrey M. Lee
Meetings: MWF, 11:30 AM to 12:20 PM in Keller 314
Instructor: Dr. George R. Wilkens
Office: Keller 308
Phone: (808) 956-4677
Hours: MT 2:30 PM to 3:30 PM, and by appointment.
E-mail: grw at math.hawaii.edu
Home Page: http://math.hawaii.edu/~grw/

Readings

The course will introduce some of the fundamental objects, maps, structures, and operators that lie at the foundations of differential geometry. These fundamentals include the idea of a smooth manifold, a smooth function between smooth manifolds, submanifolds and foliations, the tangent bundle, the cotangent bundle, and other tensor bundles that are associated to a smooth manifold. Additionally, we will introduce the notion of a vector field, its flow, and how the flow determines a notion of directional derivative (the Lie derivative). From the Lie derivative we will introduce a Lie algebra structure (the Lie bracket) on the module of smooth vector fields. Additional structures include the exterior algebra of differential forms and the exterior derivative. The final units of the course will include a brief introduction to Lie groups and Lie algebras, followed by a brief introduction to Riemannian geometry. While material will come from various parts of the text book, the bulk will be taken from Chapters 1, 2, 3, 5, and 13.

Attendance and Grading

Regular attendance to class lectures is highly recommended. The grade will be determined by homework assignments (MATH 625 Homework Page) and a take-home final exam. Homework assignments must be legible, and you are required to use proper grammatical English and follow the guidelines for clear and concise writing.


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