This course will be a continuation of Math 660 and will attempt to overview several aspects of "comparison geometry" on Riemannian manifolds, that is the study of how bounds on curvature effect the geometry and tolpology of manifolds. The first order of business will be to finish the last three chapters of Do Carmo where we will cover Preissman's Theorem and the proof of the classical homeomorphic sphere theorem.
From there the class can break into two units, the first, sectional curvature comparison will focus of manifolds with a lower sectional curvature bound. The main geometric tool here is the Toponogov triangle comparison theorem and it's main applications are the diameter sphere theorem, the soul theorem, and the betti number estimate. We should be able to cover the general aspects of this theory in the fist half of the course.
The second unit is Ricci curvature comparison. Ricci curvature is an average of sectional curvatures so a lower bound on Ricci curvature is far less restricitve then a bound on sectional curvature. Still there is something we can say about the structure of these spaces. The main tools here are the Bishop-Gromov volume comparison and the laplacian comparison theorems, as applications we will discuss the growth of fundamental groups, the first betti number estimate, and the splitting theorem.
While the above is a plan that I think willl take us through the semester there is a lot of flexibility depending on the interests of the class. For example, if the class is more interested in geometric analysis or mathematical physics than topology we could probably spend less time on the first unit of the class and perhaps also cover topics such as the Poincare inequality, isoperimetric inequalities, or harmonic functions on manifolds or even Einstein manifolds. On the other hand, if the class is interested in doing research in Riemannian geometry we should be more careful in going through the first unit in detail.
For students who may not be signed up for the class but might be interested in some topics, I will keep a calendar here of which topics we are covering as we move along.
Text: P. Petersen, Riemannian geometry, 2nd;Ed, Springer. (Optional)
Most of the topics mentioned above are covered in this book and it is a must own book if you are interested in Riemannian geometry, however you do not need to buy it and I will type up all homework assignments and post them on the webpage. There are many other books and lecture notes that cover at least some of the material mentioned above. A few of these are
1. Comparison Geometry by J.Cheeger and D. Ebin
2. Math 241 Lecture Notes by G. Wei (taken and typed by J. Ennis) which can be found on Wei's website
3. "Comparison Geometry for Ricci curvature" by S. Zhu in Comparison Geometry, MSRI,publications Vol. 30 is an excellent article about Ricci curvature and can be downloaded from MSRI's webpage.