|Lectures||MWF 1:45-2:50 (108 Kemeny)|
|X-Hour||Tu 1-1:50 (108 Kemeny)|
|Office Hours||Tu 5:30-6:30, Fri. 3:30-4:30 (also by appointment)|
|Office||321 Kemeny Hall|
|Craig.J.Sutton AT You Know Where|
|Syllabus, Announcements, etc.|
Course Description: Manifolds are one way in which mathematicians deal with the concept of "space," and the presence of a Riemannian metric on a manifold provides us with a framework and a set of tools with which we may explore the geometric and topological nature of the space in question. Among the tools available to us, curvature is perhaps the most fundamental. The three primary flavors of curvature are sectional curvature, Ricci curvature and scalar curvature, and each provides us with a local characterization of how much a given space deviates from being Euclidean. After laying the basic foundations of Riemannian geometry, one of our main objectives will be to examine how (global) constraints on curvature influence the topological and geometric structure of the space.
Topics: connections, Riemannian metrics, volume, curvature and isometries; geodesics, Jacobi fields, the energy functional, first & second variations of energy; spaces of constant curvature; (locally) symmetric spaces; isoperimetirc inequalities; spectral geometry; sphere theorems.
Prerequisites: Math 124 or the equivalent (i.e., a sound introduction to differentiable manifolds). Or, the motivtion to pick up the background material via self-study.
Textbook: Riemannian Geometry, Takashi Sakai, American Mathematical Society 1996. (available at Wheelock Books, potentially cheaper via AMS website).
Other Useful Texts (on reserve at Baker):
- A Panoramic View of Riemannian Geometry, Marcel Berger, Springer 2003. (This is a wonderful book for learning about current areas of research!)
- Riemannian Geometry (Second Edition), Peter Petersen, Springer 2006.
- Riemannian Geometry (Third Edition), S. Gallot, D. Hulin & J. Lafontaine, Springer 2004.
- Riemannian Geometry: a Modern Introduction (Second Edition), Isaac Chavel, Cambridge University Press 2006.
- Riemannian Geometry, Manfredo do Carmo, Birkhauser 1992.
(tentative) Deliverables & (tentative) Grading Guide: The following will comprise the written assignments for this term.
- Written Homework: Throughout the term you will be assigned problem sets. You are encouraged to collaborate with other memebers of the class, but your final write-up must reflect your understanding of the material and you must acknowledge the people you consulted with. No late homework will be accepted
- Oral Midterm: Scheduled Individually
- Oral Final Exam: Scheduled Individually
- Please note that the registrar has determined that February 26 is the final day to withdraw from a course this term.
Your course grade will probably be computed as follows.
Written HW 25% Midterm Exam 35% Final Exam 40%
Students with disabilities: If you have a disability and require disability related accomodations please speak to me and Ward Newmeyer, Director of Student Accessibility Services, as soon as possible so we can find a remedy.
Last Updated 2 January, 2013