Jonathan Rosenberg. You may call me at (301) 405-5166 or reach me by email at jmr@math.umd.edu. I usually attend the Geometry/Topology Seminar (Mondays 3-4), the Representation Theory and Algebra Seminars (Mondays and Wednesdays 2-3), and the Colloquium. My office hours are Mondays and Wednesdays after class, or by appointment.
MWF at 9 in MTH 1313.
MATH 405 and 411 (linear algebra and advanced calculus) or equivalent. MATH 730 is useful but not essential.
Manfredo do Carmo, Riemannian Geometry, 1992, translated by F. Flaherty, Birkhäuser, ISBN 978-0-8176-3490-2.
Riemannian geometry was created by Bernhard Riemann in his remarkable 1854 Habilitationsschrift (available at the Riemann archive as paper #13 in the original German and as paper #20 in translation). It provides the mathematical underpinnings for most analysis on manifolds as well as for general relativity theory.
This course will be a basic graduate course in differential geometry, in other words, in the use of methods of differential calculus to study manifolds. (If you don't already know what a manifold is, that's the very first topic!) Topics will include: Riemannian metrics, bundles, connections, geodesics, curvature, Jacobi fields, geometry of submanifolds. After covering all these preliminaries we will get to some of the main theorems in global Riemmannian geometry, which relate curvature properties to topology, such as the Cartan-Hadamard Theorem, Myers's Theorem, and the Sphere Theorem. If time permits we may also cover some useful topics not in the text, such as the Frobenius theorem on integrable distributions, the Soul Theorem of Cheeger-Gromoll, and the O'Neill theory of submersions.
There will be regular graded homework assignments, but no exams. If you are taking the course for credit, then you are expected to turn in the homework on time and to get a reasonable fraction of it correct.
Please go to Course Eval UM to fill in your course evaluation before December 12.
| Week | Topic | Reading Assignment | Notes |
| Aug. 27 - Aug. 31 | Smooth manifolds and tangent spaces | 0.1-0.2 | Class starts Wednesday. |
| Sept. 3 - Sept. 7 | The implicit function theorem, brackets of vector fields | 0.3-0.5 | No class Monday, Sept. 3, Labor Day. Homework #1 due Monday, Sept. 10. |
| Sept. 10 - Sept. 14 | Riemannian metrics | Ch. 1 | Homework #2 due Friday, Sept. 21. |
| Sept. 17 - Sept. 21 | Connections | Ch. 2 | No class Monday, Sept. 17,
Rosh Hashanah. Homework #3 due Friday, Sept. 28. |
| Sept. 24 - Sept. 28 | Geodesics | Ch. 3 | No class Wednesday, Sept. 26, Yom Kippur |
| Oct. 1 - Oct. 5 | The Riemann curvature | 4.1-4.3 | Homework #4 due Friday, Oct. 12. |
| Oct. 8 - Oct. 12 | Ricci, sectional, and scalar curvature | 4.4-4.5 | |
| Oct. 15 - Oct. 19 | Jacobi fields | Ch. 5 | Homework #5 due Friday, Oct. 26:
Do Carmo pp. 119-123, #1 ("normal ball" means that ε is sufficiently small), #3, #6. |
| Oct. 22 - Oct. 26 | Submanifolds and the 2nd fundamental form | Ch. 6 | Homework #6 on Ch. 6, due Friday, 11/9 Partial solutions to the homework |
| Oct. 29 - Nov. 2 | Metric and geodesic completeness | Ch. 7 | Homework #7 on Ch. 7, due Friday, 11/16 |
| Nov. 5 - Nov. 9 | Spaces of constant curvature | 8.1-8.3 | |
| Nov. 12 - Nov. 16 | Spaces of constant curvature, cont'd | 8.4-8.5 | |
| Nov. 19 - Nov. 23 | Variational formulas and Myers's Theorem | 9.1-9.2 | No class Friday, Thanksgiving Break. |
| Nov. 26 - Nov. 30 | Synge's Theorem and the Bishop-Gromov
Comparison Theorem some convenient notes by Eschenburg | 9.2-9.3 | Homework #8, due Wednesday, December 12 |
| Dec. 3 - Dec. 7 | The Bishop-Gromov and Toponogov Comparison Theorems some convenient notes by Meyer | see notes | |
| Dec. 10 - Dec. 14 | The Toponogov Comparison Theorem (cont'd) a plot to illustrate the theorem | see notes | Class ends Monday. Solutions to Homework #8 |