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Three Dimensional Manifolds All of Whose Geodesics Are Closed John Olsen Wolfgang Ziller, Advisor We present some results concerning the Morse Theory of the energy function on the free loop space of S for metrics all of whose geodesics are closed. We also show how these results may be regarded as partial results on the Berger Conjecture in dimension three.

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)

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(r) ⊂ R is such a minimal immersion, then the components of Φ must be eigenfuctions of the Laplace operator… (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 × S, also fit into this subject. For non-negatively… (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 L2 metric is precisely the negative of the… (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 Hp, i.e. all isometries of g fixing p, acts irreducibly on TpM 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)

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)

- Ricardo Mendes, Ricardo Mendes, +5 authors Lee Kennard
- 2011

EQUIVARIANT TENSORS ON POLAR MANIFOLDS Ricardo Mendes Wolfgang Ziller, Advisor This PhD dissertation has two parts, both dealing with extension questions for equivariant tensors on a polar G-manifold M with section Σ ⊂ M. Chapter 3 contains the first part, regarding the so-called smoothness conditions: If a tensor defined only along Σ is equivariant under… (More)

- Karsten Grove, Burkhard Wilking, Wolfgang Ziller, 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)

- 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)