Course Description and Grading BreakdownThe course will cover basic Riemannian geometry: geometry of Riemannian manifolds, connections, curvature, Bianchi identities, completeness, geodesics, exponential map, Gauss’s lemma, Jacobi fields, Lie groups, principal bundles, and characteristic classes. A main goal will be to prove a number of local-to-global theorems relating curvature to 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), Bonnet’s theorem (giving analogous restrictions on manifolds of strictly positive curvature), and hopefully a special case of the sphere theorem (showing that a manifold with pinched curvature is homeomorphic to a sphere). Grading: HW 50%; Final 50% Course Meeting Time and LocationMonday, Wednesday and Friday 1:00 - 1:55 pm 187 Linde Course Instructor Contact Information and Office Hourssmillie at caltech dot edu 104 Linde Hall Office Hours: Tuesday 10:30-11:30 Course Schedule and TextbookBooks used in the course: 1. Riemannian Geometry, Manfredo Do Carmo (on reserve at Sherman Fairchild Library) 2. Riemannian Manifolds: An Introduction to Curvature, John M. Lee http://caltech.tind.io/record/909376?ln=en 3. Riemannian Geometry, Peter Petersen http://caltech.tind.io/record/907769?ln=en
Course PoliciesThe due date for homework is Friday 1pm (at the start of class). Collaboration is allowed for the homework but you have to write your own solution. You are allowed to turn in one late homework (without penalty) in the quarter. However, you must inform me about turning in your homework late BEFORE the due date, and you must turn in the late homework within a week of the due date. In general, if the homework is late within a week of the due date, it is still acceptable but will be graded based on 20% points off. AssignmentsCollaboration Table
* You may use a computer or calculator while doing the homework, but may not refer to this as justification for your work. For example, "by Mathematica" is not an acceptable justification for deriving one equation from another. Also, since computers and calculators will not be allowed on the exams, it's best not to get too dependent on them. |