General Course Information, Math 547, Autumn 2011

• Instructor: Prof. Judith M. Arms, arms@math.washington.edu, C338 PDL, (206)543-9458, messages at (206)543-1150; Office hours: for the first week (or so) of classes: Immediately after class and by appointment. I will set office hours for the quarter after the class answers a survey about preferred times.


• Prerequisite: Math 544/5/6, Topology and Geometry of Manifolds; or, permission of instructor.


• Text: New draft of the second edition of Riemannian Manifolds: An Introduction to Curvature by John M. Lee.
  Electronic version (restricted access). Hard copies should be at the University Book Store now.
  Hard copies will be at the University Book Store by, hopefully, the end of September.


• Course description: Math 547 is an introduction to the primary concepts and techniques of Riemannian geometry. For the first few weeks, we will study some basic concepts (Riemannian metrics, connections, geodesics) and key examples. Chapter 6 includes our first big result, the Hopf-Rinow Theorem, which solidifies the relationship between geodesics and the topological metric derived from the Riemannian metric. In Chapter 7, we define the curvature tensor, and show it is appropriately named by proving that the curvature is zero exactly when a manifold is locally isometric to Euclidean space. Gauss's Theorema Egregium, in Chapter 8, says that the curvature of a surface in Euclidean (3 dimensional) space, defined by how it sits in space, is in fact a property of the distance function (equivalently, the Riemannian metric) on the surface.
  
  The last three chapters focus on four "fundamental theorems relating curvature and topology, ... the Gauss-Bonnet theorem (expressing the total curvature of a surface in terms of its topological type), the Cartan-Hadamard theorem (restricting the topology of manifolds of nonpositive curvature), the Bonnet-Myers theorem (giving analogous restrictions on manifolds of strictly positive curvature), and the characterization of manifolds of constant curvature." [Preface, our text.] We tackle the Gauss-Bonnet Theorem in Chapter 9. After developing yet one more tool, Jacobi fields, we will be able to discuss the other three in Chapter 11.


• Grades will be based entirely on homework; there will be no tests. I expect the cut-offs for 4.0 and 3.0 to be about 85-90% and 65-70%, respectively. For more details, see Information about homework assignments.
  
  If you wish, you may register for this course on an S/NS basis (for example, if you have passed prelims and chosen a PhD committee, and are therefore no longer required to register for graded courses). Be sure to tell me if that's what you're doing. In this case, if you attend regularly, do several reading reports, and hand in complete written solutions to at least two homework problems, I'll record your grade as a 2.7 (for grad students) or 2.0 (for undergrads), which will be converted by the registrar to S (satisfactory).

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Most recently updated on September 30, 2011.