Commutativity of the centralizer of a subalgebra in a universal enveloping algebra

Let G be a reductive algebraic group over an algebraically closed field of characteristic zero, and let \(\mathfrak{h}\) be an algebraic subalgebra of the tangent Lie algebra \(\mathfrak{g}\) of G. We find all subalgebras \(\mathfrak{h}\) that have no nontrivial characters and whose centralizers \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) and \(P(\mathfrak{g})^\mathfrak{h} \) in the universal enveloping algebra \(\mathfrak{U}(\mathfrak{g})\) and in the associated graded algebra \(P(\mathfrak{g})\), respectively, are commutative. For all these subalgebras, we prove that \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} = \mathfrak{U}(\mathfrak{h})^\mathfrak{h} \otimes \mathfrak{U}(\mathfrak{g})^\mathfrak{g} \) and \(P(\mathfrak{g})^\mathfrak{h} = P(\mathfrak{h})^\mathfrak{h} \otimes P(\mathfrak{g})^\mathfrak{g} \). Furthermore, we obtain a criterion for the commutativity of \(\mathfrak{U}(\mathfrak{g})^\mathfrak{h} \) in terms of representation theory.     

Orbifold fibrations of Eschenburg spaces

Most of the few known examples of compact Riemannian manifolds with positive sectional curvature are the total space of a Riemannian submersion. In this article we show that this is true for all known examples, if we enlarge the category to orbifold fibrations. For this purpose we study all almost free isometric circle actions on positively curved Eschenburg spaces, which give rise to principle orbifold bundle structures, and we examine in detail their geometric properties. In particular, we obtain a new family of 6-dimensional orbifolds with positive sectional curvature whose singular locus consists of just two points.      

Pseudofree -actions on surfaces

In this paper, we give a weak classification of locally linear pseudofree actions of the cyclic group of order on a surface, and prove the existence of such an action which cannot be realized as a smooth action on the standard smooth surface.

     

Rigidity of higher rank abelian cocycles with values in diffeomorphism groups

We consider cocycles over certain hyperbolic actions, , and show rigidity properties for cocycles with values in a Lie group or a diffeomorphism group, which are close to identity on a set of generators, and are sufficiently smooth. The actions we consider are Cartan actions of or , for , and Γ torsion free cocompact lattice. The results in this paper rely on a technique developed recently by D. Damjanović and A. Katok.      

The biinvariant diagonal class for Hamiltonian torus actions

Suppose that an algebraic torus G acts algebraically on a projective manifold X with generically trivial stabilizers. Then the Zariski closure of the set of pairs {(x,y)∈X×X|y=gx for some g∈G} defines a nonzero equivariant cohomology class . We give an analogue of this construction in the case where X is a compact symplectic manifold endowed with a Hamiltonian action of a torus, whose complexification plays the role of G. We also prove that the Kirwan map sends the class [ΔG] to the class of the diagonal in each symplectic quotient. This allows to define a canonical right inverse of the Kirwan map.     

Invariant Metrics on Cohomogeneity One Manifolds

Let G be one of the connected subgroups of the orthogonal group of ⁿ which acts transitively on the unit sphere S ^n–1. We get the necessary and sufficient condition for G-invariant metrics g on ⁿ \{0} to be extendend to the origin. For n=2 this is a classical result of Berard–Bergery. The curvature tensor and the sectional curvature of any such Riemannian G-manifold ( ⁿ , g) are described in terms of the length of the Killing vector fields, as well as the second fundamental form of the regular orbits G(P)=S ^n–1. As an application we describe all G-invariant metrics which are Kähler, hyperKähler or have constant principal curvatures. Some of these results are generalized to the case of any cohomogeneity one G-manifold which, in a neighbourhood of a singular orbit, can be identified with a twisted product.     

Invariant discrete flows

In this paper, we investigate the evolution of joint invariants under invariant geometric flows using the theory of equivariant moving frames and the induced invariant discrete variational complex. For certain arc length preserving planar curve flows invariant under the special Euclidean group , the special linear group , and the semidirect group , we find that the induced evolution of the discrete curvature satisfies the differential‐difference mKdV, KdV, and Burgers' equations, respectively. These three equations are completely integrable, and we show that a recursion operator can be constructed by precomposing the characteristic operator of the curvature by a certain invariant difference operator. Finally, we derive the constraint for the integrability of the discrete curvature evolution to lift to the evolution of the discrete curve itself.     

When is a Riemannian submersion homogeneous?

We study the structure of the most common type of Riemannian submersions, namely those whose fibers are given by the orbits of an isometric group action on a Riemannian manifold. Special emphasis is given to the case where the ambient space has nonnegative curvature. The first author was supported by research grant MTM2004-04794-MEC. Most of this work was done during a visit of the second author to Madrid, financed in part by funds of the aforementioned grant.     

Growth and homogeneity of matchbox manifolds

A matchbox manifold with one-dimensional leaves which has equicontinuous holonomy dynamics must be a homogeneous space, and so must be homeomorphic to a classical Vietoris solenoid. In this work, we consider the problem, what can be said about a matchbox manifold with equicontinuous holonomy dynamics, and all of whose leaves have at most polynomial growth type? We show that such a space must have a finite covering for which the global holonomy group of its foliation is nilpotent. As a consequence, we show that if the growth type of the leaves is polynomial of degree at most 3, then there exists a finite covering which is homogeneous. If the growth type of the leaves is polynomial of degree at least 4, then there are additional obstructions to homogeneity, which arise from the structure of nilpotent groups.