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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)

- Karsten Grove, Wolfgang Ziller
- 2007

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)

- Vitali Kapovitch, Wolfgang Ziller
- 2002

We classify all biquotients whose rational cohomology rings are generated by one element. As a consequence we show that the Gromoll-Meyer 7-sphere is the only exotic sphere which can be written as a biquotient. on the left by the formula (h 1 , h 2)g = h 1 gh −1 2. If this action happens to be free the orbit space is a manifold which is called a biquotient… (More)

- Dennis Deturck, Wolfgang Ziller
- 2007

Introduction. A number of authors [C], [DW1], [DW2], [L], [T] have studied minimal isometric immersions of Riemannian manifolds into round spheres, and in particular of round spheres into round spheres. As was observed by T. Takahashi [T], if Φ: M → S N (r) ⊂ R N +1 is such a minimal immersion, then the components of Φ must be eigenfuctions of the Laplace… (More)

- Karsten Grove, Burkhard Wilking, Wolfgang Ziller
- 2005

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)

- Wolfgang Ziller
- 2007

Manifolds with non-negative sectional curvature have been of interest since the beginning of global Riemannian geometry, as illustrated by the theorems of Bonnet-Myers, Synge, and the sphere theorem. Some of the oldest conjectures in global Riemannian geometry, as for example the Hopf conjecture on S 2 × S 2 , also fit into this subject. For non-negatively… (More)

- Karsten Grove, Wolfgang Ziller
- 2007

One of the central problems in Riemannian geometry is to determine how large the classes of manifolds with positive/nonnegative sectional-, Ricci-or scalar curvature are (see [Gr]). For scalar curvature the situation is fairly well understood by comparison. Special surgery constructions as in [SY, Wr] and bundle constructions as in [Na] have resulted in a… (More)

- Ted Chinburg, Christine Escher, Wolfgang Ziller, T Chinburg, W Ziller, W Ziller +1 other
- 2005

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)

- Karsten Grove, Luigi Verdiani, Burkhard Wilking, Wolfgang Ziller
- 2006

Most examples of manifolds with non-negative sectional curvature are obtained via product and quotient constructions, starting from compact Lie groups with bi-invariant metrics. In [GZ1] a new large class of non-negatively curved compact manifolds was constructed by using Lie group actions whose quotients are one-dimensional, so called cohomogeneity one… (More)

- Karsten Grove, Wolfgang Ziller
- 2008

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)