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Karsten Grove

Publications and source records attributed to Karsten Grove.

16 recordsLinked to original sources
arXiv2018

Alexandrov Spaces with Maximal Radius

Abstract. In this paper we prove several rigidity theorems related to and including Lytchak's problem. The focus is on Alexandrov spaces with \curv\geq1, nonempty boundary, and maximal radius \frac{\pi}{2}. We exhibit many such spaces that indicate that this class is remarkably flexible. Nevertheless, we also show that when the boundary is either geometrically or topologically spherical, then it is possible to obtain strong rigidity results. In contrast to this one can show that with general lower curvature bounds and strictly convex boundary only cones can have maximal radius. We also mention some connections between our problems and the positive mass conjectures. This paper is an expanded version and replacement of the two previous versions

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arXiv2018

The Boundary Conjecture for Leaf Spaces

We prove that the boundary of an orbit space or more generally a leaf space of a singular Riemannian foliation is an Alexandrov space in its intrinsic metric, and that its lower curvature bound is that of the leaf space. A rigidity theorem for positively curved leaf spaces with maximal boundary volume is also established and plays a key role in the proof of the boundary problem.

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arXiv2016

Rank three geometry and positive curvature

An axiomatic characterization of buildings of type $\CC_3$ due to Tits is used to prove that any cohomogeneity two polar action of type $\CC_3$ on a positively curved simply connected manifold is equivariantly diffeomorphic to a polar action on a rank one symmetric space. This includes two actions on the Cayley plane whose associated $\CC_3$ type geometry is not covered by a building.

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arXiv2014

Rigidity theorems for submetries in positive curvature

We derive general structure and rigidity theorems for submetries $f: M \to X$, where $M$ is a Riemannian manifold with sectional curvature $\sec M \ge 1$. When applied to a non-trivial Riemannian submersion, it follows that $diam X \leq π/2 $. In case of equality, there is a Riemannian submersion $\mathbb{S} \to M$ from a unit sphere, and as a consequence, $f$ is known up to metric congruence. A similar rigidity theorem also holds in the general context of Riemannian foliations.

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arXiv2014

Reflection groups in non-negative curvature

We provide an equivariant description/classification of all complete (compact or not) non-negatively curved manifolds M together with a co-compact action by a reflection group W, and moreover, classify such W. In particular, we show that the building blocks consist of the classical constant curvature models and generalized open books with non negatively curved bundle pages, and derive a corresponding splitting theorem for the universal cover.

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arXiv2013

A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetry

We classify nonnegatively curved simply connected 4-manifolds with circle symmetry up to equivariant diffeomorphisms. The main problem is rule out knotted curves in the singular set of the orbit space. As an extension of this work we classify all knots in S^3 which can be realized as an extremal set with respect to an inner metric on S^3 which has nonnegative curvature in the Alexandrov sense.

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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.

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arXiv2012

Global G-Manifold reductions and resolutions

The purpose of this note is to exhibit some simple and basic constructions for smooth compact transformation groups, and some of their most immediate applications to geometry.

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arXiv2012

Tits Geometry and Positive Curvature

There is a well known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible topological spherical buildings of rank at least three. We discover and exploit a rich structure of a (connected) chamber system of finite (Coxeter) type M associated with any polar action of cohomogeneity at least two on any simply connected closed positively curved manifold. Although this chamber system is typically not a Tits geometry of type M, we prove that in all cases but two that its universal Tits cover indeed is a building. We construct a topology on this universal cover making it into a compact spherical building in the sense of Burns and Spatzier. Using this structure we classify up to equivariant diffeomorphism all polar actions on (simply connected) positively curved manifolds of cohomogeneity at least two.

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arXiv2009

Developments around positive sectional curvature

This is not in any way meant to be a complete survey on positive curvature. Rather it is a short essay on the fascinating changes in the landscape surrounding positive curvature. In particular, details and many results and references are not included, and things are not presented in chronological order.

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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.

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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.

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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.

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arXiv1992

Curvature, triameter, and beyond

In its most general form, the recognition problem in Riemannian geometry asks for the identification of an unknown Riemannian manifold via measurements of metric invariants on the manifold. We introduce a new infinite sequence of invariants, the first term of which is the usual diameter, and illustrate the role of these global shape invariants in a number of recognition problems.

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