Free, Isometric Circle Actions on Compact Symmetric Spaces

Let S=G/K be a strongly irreducible, simply connected, compact symmetric space and let be its group of isometries. We classify the symmetric spaces among these that admit free, isometric circle actions. The existence of such actions is important in constructing examples of manifolds with positive sectional curvature.     

Nonnegatively curved homogeneous metrics obtained by scaling fibers of submersions

We consider invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate subgroup K between G and H exists, so that the homogeneous space G/H is the total space of a Riemannian submersion. We study the question as to whether enlarging the fibers of the submersion by a constant scaling factor retains the nonnegative curvature in the case that the deformation starts at a normal homogeneous metric. We classify triples of groups (H, K, G) where nonnegative curvature is maintained for small deformations, using a criterion proved by Schwachhöfer and Tapp. We obtain a complete classification in case the subgroup H has full rank and an almost complete classification in the case of regular subgroups.     

Low cohomogeneity and polar actions on exceptional compact Lie groups

We study isometric Lie group actions on the compact exceptional groups E₆, E₇, E₈, F₄ and G₂ endowed with a bi-invariant metric. We classify polar actions on these groups, in particular, we show that all polar actions are hyperpolar. We determine all isometric actions of cohomogeneity less than three on E₆, E₇, F₄ and all isometric actions of cohomogeneity less than 20 on E₈. Moreover, we determine the principal isotropy algebras for all isometric actions on G₂.     

Homogeneous Spaces with Sections

We study homogeneous Riemannian manifolds all of whose geodesics can be mapped by some isometry into a fixed homogeneous, connected, totally geodesic submanifold, called section. We show that these spaces are locally symmetric if the section is two-dimensional and give non-symmetric counterexamples with higher-dimensional sections.Mathematics Subject Classification (2000): 53C35, 53C30, 53C22, 57S15     

A classification of hyperpolar and cohomogeneity one actions

An isometric action of a compact Lie group on a Riemannian manifold is called hyperpolar if there exists a closed, connected submanifold that is flat in the induced metric and meets all orbits orthogonally. In this article, a classification of hyperpolar actions on the irreducible Riemannian symmetric spaces of compact type is given. Since on these symmetric spaces actions of cohomogeneity one are hyperpolar, i.e. normal geodesics are closed, we obtain a classification of the homogeneous hypersurfaces in these spaces by computing the cohomogeneity for all hyperpolar actions. This result implies a classification of the cohomogeneity one actions on compact strongly isotropy irreducible homogeneous spaces.

     

Low Cohomogeneity Representations and Orbit Maximal Actions

An orthogonal representation of a compact Lie group is called polar if thereexists a linear subspace which meets all orbits orthogonally.It has been shown by Conlon that one can associate a Coxeter groupto such a representation.From this, an upper bound for the cohomogeneity of an irreduciblepolar representation can be derived.Another property of irreducible polar representations isthat the action restricted to the unit spherehas maximal orbits in the sense that any action having largerorbits is transitive.We give a classification of orbit maximal actions on spheresand use it to show that irreducible polar representations arecharacterized by these two properties.     

Homogeneous spaces with polar isotropy

 We consider homogeneous spaces G/K with G a simple compact Lie group, endowed with an arbitrary G-invariant Riemannian metric. We classify those spaces where the action of K on G/K is polar and show that such spaces are locally symmetric. Moreover we give a classification of pairs (G,K) with G compact semisimple such that K has polar linear isotropy representation. Received: 16 May 2002 / Revised version: 8 November 2002 Published online: 3 March 2003 Mathematics Subject Classification (2000): 53C35, 57S15     

Nonnegatively curved homogeneous metrics in low dimensions

We consider invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate subgroup K between G and H exists. In this case, the homogeneous space G/H is the total space of a Riemannian submersion. The metrics constructed by shrinking the fibers in this way can be interpreted as metrics obtained from a Cheeger deformation and are thus well known to be nonnegatively curved. On the other hand, if the fibers are homothetically enlarged, it depends on the triple of groups (H, K, G) whether non-negative curvature is maintained for small deformations. Building on the work of Schwachhöfer and Tapp (J. Geom. Anal. 19(4):929–943, 2009), we examine all G-invariant fibration metrics on G/H for G a compact simple Lie group of dimension up to 15. An analysis of the low dimensional examples provides insight into the algebraic criteria that yield continuous families of non-negative sectional curvature.     

Codimension of immersions with parallel pluri-mean curvature

Immersions with parallel pluri-mean curvature into euclidean n-space generalize constant mean curvature immersions of surfaces to Kähler manifolds of complex dimension m. Examples are the standard embeddings of Kähler symmetric spaces into the Lie algebra of its transvection group. We give a lower bound for the codimension of arbitrary ppmc immersions. In particular we show that M is locally symmetric if the codimension is minimal.