Transitive Actions on Lorentz Manifolds with Noncompact Stabilizer

If a topological group G acts on a topological space X, then we say that the action is orbit nonproper provided that, for some xX, the orbit map ggx:GX is nonproper. We consider the problem of classifying the connected, simply connected real Lie groups G admitting a locally faithful, orbit nonproper, isometric action on a connected Lorentz manifold. In an earlier paper, we found three collections of groups such that G admits such an action iff G is in one of the three collections. In another paper, we effectively described the first collection. In this paper, we show that the second collection contains a small, effectively described collection of groups, and, aside from those, it is contained in the union of the first and third collections. Finally, in a third paper, we effectively describe the third collection, thus solving the stated problem.     

Dynamics of Simple Lie Groups on Lorentz Manifolds

We give an alternate proof of N. Kowalsky's theorem describing the collection of connected simple Lie groups with finite center which admit a nontrivial, nonproper action by isometries of a connected Lorentz manifold.     

Conformal actions of on Lorentz manifolds

We prove that, for , a locally faithful action of or of by conformal transformations of a connected Lorentz manifold must be a proper action.

     

On Lorentz dynamics: From group actions to warped products via homogeneous spaces

We show a geometric rigidity of isometric actions of non-compact (semisimple) Lie groups on Lorentz manifolds. Namely, we show that the manifold has a warped product structure of a Lorentz manifold with constant curvature by a Riemannian manifold.

     

Homogeneity Rank and Atoms of Actions

We introduce two related concepts for smooth actions of compact Lie groups:The homogeneity rank is a simple numerical invariant of the action.As one of our results we determine the precise range of this invariantfor isometric actions on compact Riemannian manifolds with positivesectional curvature and exhibit special properties of the actionswith maximal homogeneity rank.Atoms are special components of fixed point sets. They inherit actionswith the same cohomogenity and homogeneity rank as the original action,but with trivial principal isotropy group. Other properties of the originalaction like polarity are reflected in the atoms.We determine the atoms in some interesting concrete cases.Not only for this purpose we give a detailed treatise on the structureof fixed point sets, in particular in cohomogeneity one manifolds.     

On isometries of Kähler manifolds

Summary We show that an isometry between not locally hyperk?hler, locally irreducible K?hler manifolds is either holomorphic or antiholomorphic.     

Fundamental Groups of Closed Manifolds with Positive Curvature and Torus Actions

We determine the fundamental group of a closed n-manifold of positive sectional curvature on which a torus T^k (k large) acts effectively and isometrically. Our results are: (A) If k>(n − 3)/4 and n ≥ 17, then the fundamental group π₁(M) is isomorphic to the fundamental group of a spherical 3-space form. (B) If k ≥ (n/6)+1 and n≠ 11, 15, 23, then any abelian subgroup of π₁(M) is cyclic. Moreover, if the T^k-fixed point set is empty, then π₁(M) is isomorphic to the fundamental group of a spherical 3-space form.Mathematics Subject Classification (2000). 53-XX*Supported partially by NSF Grant DMS 0203164 and by a reach found from Beijing normal university.**Supported partially by NSFC 10371008.     

On Isometric Actions of SL(n,Z) on Three-Dimensional Visibility Manifolds

Let X be a three-dimensional, visibility manifold. Suppose L(n,Z), n3 acts on X by isometries. Then either the action is trivial or a point in the boundary of X is fixed by the action.     

Complete Flat Affine and Lorentzian Manifolds

Following in the tradition of Hilbert's 18th problem of classifying crystallographic groups, we provide a survey of a series of results which have culminated in the study of flat Lorentz manifolds. In particular, Milnor asked whether all complete flat affine manifolds have virtually polycyclic fundamental groups. Margulis answered this question negatively by constructing complete flat Lorentz manifolds with free fundamental groups. In this paper, we follow the effort to classify and understand these interesting counterexamples to Milnor's question, and their generalizations.     

Globalization of Holomorphic Actions on Principal Bundles

Suppose G is a Lie group acting as a group of holomorphic automorphisms on a holomorphic principal bundle P → X. We show that if there is a holomorphic action of the complexification G^C of G on. X, this lifts to a holomorphic action of G^C on the bundle P → X. Two applications are presented. We prove that given any connected homogeneous complex manifold G/H with more than one end, the complexification G^C of G acts holomorphically and transitively on G/H. We also show that the ends of a homogeneous complex manifold G/H with more than two ends essentially come from a space of the form S/Γ, where Γ is a Zariski dense discrete subgroup of a semisimple complex Lie group S with S and Γ being explicitly constructed in terms of G and H.     

Affine Deformations of Ultraideal Triangle Groups

In this paper, we give conditions for an affine ultraideal triangle group to act properly discontinuously on affine three-space.     

Some Groups Having Only Elementary Actions on Metric Spaces with Hyperbolic Boundaries

We study isometric actions of certain groups on metric spaces with hyperbolic-type bordifications. The class of groups considered includes SL _n (), Artin braid groups and mapping class groups of surfaces (except the lower rank ones). We prove that in various ways such actions must be elementary. Most of our results hold for non-locally compact spaces and extend what is known for actions on proper CAT(-1) and Gromov hyperbolic spaces. We also show that SL _n () for n 3 cannot act on a visibility space X without fixing a point in . Corollaries concern Floyd's group completion, linear actions on strictly convex cones, and metrics on the moduli spaces of compact Riemann surfaces. Some remarks on bounded generation are also included.     

Bordism-Finiteness and Semi-simple Group Actions

We give bordism-finiteness results for smooth S ³-manifolds. Consider the class of oriented manifolds which admit an S ¹-action with isolated fixed points such that the action extends to an S ³-action with fixed point. We exhibit various subclasses, characterized by an upper bound for the Euler characteristic and properties of the first Pontryagin class p ₁, for example p ₁ = 0, which contain only finitely many oriented bordism types in any given dimension. Also we show finiteness results for homotopy complex projective spaces and complete intersections with S ³-action as above.     

Geometric properties of almost contact metric 3-submersions

Summary In this paper, we discuss some geometric properties of three types of Riemannian submersions whose total space is an almost contact metric manifold with 3-structure. The study is focused on the transference of structures.     

Cohomogeneity Two Actions on Flat Riemannian Manifolds

In this paper, we study flat Riemannian manifolds which have codimension two orbits, under the action of a closed and connected Lie group G of isometries. We assume that G has fixed points, then characterize M and orbits of M.     

Isometric Actions of Heisenberg Groups on Compact Lorentz Manifolds

We prove results toward classifying compact Lorentz manifolds on which Heisenberg groups act isometrically. We give a general construction, leading to a new example, of codimension-one actions – those for which the dimension of the Heisenberg group is one less than the dimension of the manifold. The main result is a classification of codimension-one actions, under the assumption they are real-analytic.     

Compact Lorentzian holonomy

We consider (compact or noncompact) Lorentzian manifolds whose holonomy group has compact closure. This property is equivalent to admitting a parallel timelike vector field. We give some applications and derive some properties of the space of all such metrics on a given manifold.     

Translating Surfaces of the Non-parametric Mean Curvature Flow in Lorentz Manifold M2×R

In this paper, for the Lorentz manifold M²× R with M² a 2-dimensional complete surface with nonnegative Gaussian curvature, the authors investigate its spacelike graphs over compact, strictly convex domains in M², which are evolving by the nonparametric mean curvature flow with prescribed contact angle boundary condition, and show that solutions converge to ones moving only by translation.     

Residual properties of graph manifold groups

Let f:M→N be a continuous map between closed irreducible graph manifolds with infinite fundamental group. Perron and Shalen (1999) [16] showed that if f induces a homology equivalence on all finite covers, then f is in fact homotopic to a homeomorphism. Their proof used the statement that every graph manifold is finitely covered by a 3-manifold whose fundamental group is residually p for every prime p. We will show that this statement regarding graph manifold groups is not true in general, but we will show how to modify the argument of Perron and Shalen to recover their main result. As a by-product we will determine all semidirect products Z?Zd which are residually p for every prime p.     

A Remark on a Theorem by Kodama and Shimizu

We prove a characterization theorem for the unit polydisc Δ ⁿ ⊂ℂ ⁿ in the spirit of a recent result due to Kodama and Shimizu. We show that if M is a connected n-dimensional complex manifold such that (i) the group Aut (M) of holomorphic automorphisms of M acts on M with compact isotropy subgroups, and (ii) Aut (M) and Aut (Δ ⁿ ) are isomorphic as topological groups equipped with the compact-open topology, then M is holomorphically equivalent to Δ ⁿ .