Wolfgang Ziller

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1 ACKNOWLEDGEMENTS I would like to thank some of the many people who have helped me along: I thank David Harbater, Steve Shatz, and Frank Warner for encouragement throughout my graduate studies. I also thank Chris Croke and Herman Gluck for generously sharing their expertise. I am especially indebted to my advisor, Wolfgang Ziller, for so many hours of his(More)
Since Milnor's discovery of exotic spheres [Mi], one of the most intriguing problems in Rie-mannian geometry has been whether there are exotic spheres with positive curvature. It is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature [Hi]. On the other hand, there are many examples of exotic spheres with(More)
Einstein metrics of volume 1 on a closed manifold can be characterized variationally as the critical points of the Hilbert action [Hi], which associates to each Riemannian metric of volume 1 the integral of its scalar curvature. As is well-known, the gradient vector of the Hilbert action with respect to the natural L 2 metric is precisely the negative of(More)
We provide an exhaustive description of all simply connected positively curved cohomogeneity one manifolds. The description yields a complete classification in all dimensions but 7, where in addition to known examples, our list contains one exceptional space and two infinite families not yet known to carry metrics of positive curvature. The infinite(More)
We examine topological properties of the seven-dimensional positively curved Eschenburg biquotients and find many examples which are homeomorphic but not diffeomorphic. A special subfamily of these manifolds also carries a 3-Sasakian metric. Among these we construct a pair of 3-Sasakian spaces which are diffeomorphic to each other, thus giving rise to the(More)
Since the emergence of the fundamental structure theorem for complete open manifolds of nonnegative curvature due to Cheeger and Gromoll [CG], one of the central issues in this area has been to what extent the converse to this so-called soul theorem holds. In other words: Which total spaces of vector bundles over compact nonnegatively curved manifolds admit(More)
Compact manifolds that admit a metric with positive sectional curvature are still poorly understood. In particular, there are few known obstructions for the existence of such metrics. By Bonnet-Meyers the fundamental group must be finite, by Synge it has to be 0 or Z 2 in even dimensions, and thê A-genus must vanish when the manifold is spin. For(More)