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Wolfgang Ziller

Publications and source records attributed to Wolfgang Ziller.

34 records · Page 2Linked to original sources
arXiv2012

Polar manifolds and actions

A group action is called polar if there exists an immersed submanifold (a section) which intersects all orbits orthogonally. Such group actions have been studied extensively on symmetric spaces. We show how to construct a manifold admitting a polar group action by prescribing their isotropy groups along a fundamental domain, generalizing the classical construction for cohomogeneity manifolds. We give many examples showing the richness of this class of group actions. We also relate the topology of the section to the topology of the manifold. This is a replacement of an earlier version. Small changes, and a correction in Lemma 2.4.

math.DG↗
arXiv2012

Topology of non-negatively curved manifolds

We examine several classes of manifolds which have the same cohomology ring as an Eschenburg space (a family of biquotients which is a main source of manifolds with positive curvature). One family are the 3-sphere bundles over CP^2. Another are the circle bundles over a base, which itself is one of the family of CP^1 bundles over CP^2. We classify such manifolds up to diffeomorphism using the Kreck-Stolz invariants. Comparisons of the invariants is then used to find many diffeomorphism of the total space of these bundles with positively curved Eschenburg spaces. The total space of each bundle (in the case where the bundle is not spin) also admit a metric with non-negative sectional curvature, and in some cases an Einstein metric as well.

math.DG↗
arXiv2012

Concavity and rigidity in non-negative curvature

This is a significantly improved version with new applications. We show that there are many cohomogeneity one manifolds which do not admit an analytic invariant metric with non-negative sectional curvature, although they do have a smooth one. In particular, there are no invariant metric with positive curvature.

math.DG↗
arXiv2011

Topological obstructions to fatness

Alan Weinstein showed that certain characteristic numbers of any Riemannian submersion with totally geodesic fibers and positive vertizontal curvatures are nonzero. In this paper we explicitly compute these invariants in terms of Chern and Pontrjagin numbers of the bundle. This allows us to show that many bundles do not admit such metrics.

math.DG↗
arXiv2009

On M. Mueter's Ph.D. Thesis on Cheeger deformations

These are notes of a talk I gave in a seminar at the University of Pennsylvania summarizing results in the Ph.D. thesis of Michael Mueter obtained under the direction of Wolfgang Meyer at the University of Muenster. His thesis on "Kruemmungserhoehende Deformationen mittels Gruppenaktionen" examines in detail curvature properties of so called Cheeger deformations. Such deformations have been a crucial ingredient in non-negative and positive curvature constructions. His thesis contains a wealth of interesting properties of such deformations, but since it was never published, is only known to a few experts. I have no intention to publish these notes, but post them as a service to the public. A scanned version of his thesis (written in German) is available on my homepage.

math.DG↗
arXiv2009

On Eschenburg's Habilitation on Biquotients

These are notes of a talk I gave in a seminar at the University of Pennsylvania summarizing results in the Habilitation by Jost Eschenburg on "Freie isometrische Aktionen auf kompakten Lie-Gruppen mit positiv gekruemmten Orbitraeumen". Due to the fact that it is published in a not easily accesible journal (and is written in German) some of the results of his Habilitation are not as well known as they deserve. Although the main results on new examples with positive curvature were later published elsewhere, it contains a wealth of further information about biquotients and a classification of equal rank biquotients of simple Lie groups. I have no intention to publish these notes, but post them as a service to the public. A scanned version of his Habilitation (written in German) is available on my homepage.

math.DG↗
arXiv2008

Lifting Group Actions and Nonnegative Curvature

We show that all vector bundles over CP^2 which are not spin admit a complete metric with nonnegative sectional curvature. In the proof we construct a nonnegatively curved metric on the corresponding principle bundle by showing that it admits a cohomogeneity one action with singular orbits of codimension 2. This is closely related to the problem of when an action of G on the base of an L principle bundle lifts to the total space, such that the lift commutes with L. We solve this lifting problem for all SO(k) principle bundles over a 4-dimensional simply connected base B with G a cohomogeneity one action on B.

math.DG↗
arXiv2007

On the geometry of cohomogeneity one manifolds with positive curvature

This is a survey on cohomogeneity one manifolds with positive curvature. We discuss the known examples of this type and their geometry and the functions that describe the metric. We also describe the classification of cohomogeneity one manifolds that can admit a metric with positive curvature due to Grove-Wilking-Ziller. Two series of candidates arose in this classification for which it is not yet know if they admit positive curvature. The connection of these candidates to the Hitchin self dual Einstein orbifolds is discussed as well, including the curvature properties of the Hitchin metrics.

math.DG↗
arXiv2007

Positively curved homogeneous metrics on spheres

We examine homogeneous metrics on spheres and determine which ones have positive sectional curvature. The answer is subtle and surprisingly difficult to prove. In some cases we also determine their pinching constants. This completes the classification of all homogeneous metrics with positive curvature (apart from one special Aloff Wallach space where the set of homogeneous metrics is too large).

math.DG↗
arXiv2006

Orbifold fibrations of Eschenburg spaces

We study isometric circle actions on 7 dimensional positively curved Eschenburg spaces which are almost free, thus giving rise to orbifold fibrations of these spaces. This shows in particular that every known example of compact manifolds with positive sectional curvature is the total space of an orbifold fibration. We then study in detail the geometry of the singular locus and how to minimize it. As a consequence we obtain, for example, a six dimensional compact positively curved orbifold with only two singular points with orbifold groups Z_2.

math.DG↗
arXiv2006

Symmetries of Eschenburg spaces and the Chern problem

The known manifolds of positive sectional curvature are either homogeneous spaces or biquotients, i.e. quotients of a compact Lie group by a group acting on the left and right simultaneously. The full isometry group of the homogeneous metrics of positive curvature were determined by K.Shankar. Here we determine the isometry group of some of the biquotients due to Eschenburg and Bazaikin. As an application we obtain, as in the homogeneous case, more counterexamples to the Chern conjecture, which states that an abelian subgroup of the fundamental group of a positively curved manifold is cyclic.

math.DG↗
arXiv2006

On the topology of positively curved Bazaikin spaces

We study the topology of the 13 dimensional positively curved Bazaikin spaces. We show that there is only one such manifold which is homotopy equivalent to a homogeneous space, the so called Berger space. This is in contrast to the case of the 7 dimensional positively curved Eschenburg spaces. In addition, we compute the Pontryagin classes and the linking form and show that the first two billion positively curved Bazaikin manifolds are homeomorphically distinct, raising the question whether this is true in general.

math.DG↗
arXiv2004

Topological properties of Eschenburg spaces and 3-Sasakian manifolds

The authors examine topological properties of the 7-dimensional Eschenburg biquotients diag(z^k1,z^k2,z^k3)\SU(3)/diag(z^l1,z^l2,z^l3). A subfamily of these spaces carry a 3-Sasakian metric. The authors show that among this subfamily there exist many 3-Sasakian spaces which are homeomorphic but not diffeomorphic. In addition, they construct a pair of 3-Sasakian spaces which are diffeomorphic, thus giving the first example of a manifold which carries two non-isometric 3-Sasakian metrics. This answers an open question of C. Boyer and K.Galicki. Among the general family, the authors construct many pairs of positively curved Eschenburg spaces which are homeomorphic but not diffeomorphic. Such pairs were first constructed by Kreck-Stolz among the special subfamily of Aloff-Wallach spaces.

math.DG↗
arXiv2000

Curvature and symmetry of Milnor spheres

In this paper we explore the geometry and topology of cohomogeneity one manifolds, i.e. manifolds with a group action whose principal orbits are hypersurfaces. We show that the principal group action of every principal SO(3) and SO(4) bundle over S^4 extends to a cohomogeneity one action. As a consequence we prove that every vector bundle and every sphere bundle over S^4 has a complete metric with non-negative curvature. It is well known that 15 of the 27 exotic spheres in dimension 7 can be written as S^3 bundles over S^4 in infinitely many ways, and hence we obtain infinitely many non-negatively curved metrics on these exotic spheres. A further consequence will be that there are infinitely many almost free actions by SO(3) on S^7, i.e. all isotropy groups are finite. These actions preserve the Hopf fibration S^3 -> S^7 -> S^4 but do not extend to the disc D^8. We also construct infinitely many such actions on the 15 exotic 7-spheres mentioned above.

math.DG↗