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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)

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)

- WOLFGANG ZILLER
- 1991

A connected Riemannian manifold (M, g) is said to be isotropy irreducible if for each point p ∈ M the isotropy group H p , i.e. all isometries of g fixing p, acts irreducibly on T p M via its isotropy representation. This class of manifolds is of great interest since they have a number of geometric properties which follow immediately from the definition. By… (More)

- Karsten Grove, Burkhard Wilking, Wolfgang Ziller, W. 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)

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)

A strongly isotropy irreducible homogeneous space is a connected effective homogeneous space L/H where H is a compact subgroup of a Lie group L such that the identity component H 0 of H acts irreducibly on T eH (L/H). If we assume instead that H, but not necessarily H 0 , acts irreducibly on T eH (L/H), then L/H is called an isotropy irreducible homogeneous… (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)