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Group Actions on Positively Curved Manifolds Dr. K. Shankar University of Michigan, U.S.A. 10-08-01 Abstract A general problem in Riemannian geometry is to find and describe manifolds $M^n$ that admit a complete Riemannian metric of positive sectional curvature. If the curvature is allowed to go to zero asymptotically, then the Cheeger-Gromoll-Meyer Soul theorem tells us that the manifold is diffeomorphic to ${\Bbb R}^n$. When there is a positive lower bound on the sectional curvature, the Bonnet-Myers and Synge theorems say that $M^n$ must be compact and $\pi_1(M)$ is finite. However, there are few known obstructions and the known examples, although infinite, are relatively few in number. In this talk we study the known examples and present counterexamples to an old conjecture of S.S. Chern about fundamental groups of positively curved manifolds. Chern asked: if $M^n$ is a positively curved manifold, then is it true that every abelian subgroup of $\pi_1(M^n)$ is cyclic? We will see the motivation behind this conjecture and then construct counterexamples. Most of the background required will be presented in the talk. Some basic knowledge about manifolds, fundamental groups, Riemannian metrics and Lie groups will be assumed.
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