Quotients Of Manifolds Research Articles

A Boothby\u2013Wang Theorem for Besse Contact Manifolds

A Reeb flow on a contact manifold is called Besse if all its orbits are periodic, possibly with different periods. We characterize contact manifolds whose Reeb flows are Besse as principal S^1-orbibundles over integral symplectic orbifolds satisfying some cohomological condition. Apart from the cohomological condition, this statement appears in the work of Boyer and Galicki in the language of Sasakian geometry (Boyer and Galicki in Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008). We illustrate some non-commonly dealt with perspective on orbifolds in a proof of the above result. More precisely, we work with orbifolds as quotients of manifolds by smooth Lie group actions with finite stabilizer groups. By introducing all relevant orbifold notions in this equivariant way, we avoid patching constructions with orbifold charts. As an application, and building on work by Cristofaro-Gardiner–Mazzucchelli, we deduce a complete classification of closed Besse contact 3-manifolds up to strict contactomorphism.

Quotients of projective spaces and spheres

Abstract In this paper we prove results on topological properties of quotients of manifolds, in particular of projective spaces and spheres. Specifically, under suitable conditions we exhibit results on homology and homotopy groups of these quotients. As an application we generalize the characterization of topological translation planes obtained by Löwen, Steinke, and Van Maldeghem in [Adv. Geom.: S59–S74, 2003] to all possible (finite) dimensions. Our approach also yields a more unified proof for this characterization result. Most of the results in this paper are obtained by means of standard results from algebraic topology.

Stringy Chern classes of singular varieties

Motivic integration [M. Kontsevich, Motivic integration, Lecture at Orsay, 1995] and MacPherson's transformation [R. MacPherson, Chern classes for singular varieties, Ann. of Math. 100 (1974) 423–432] are combined in this paper to construct a theory of “stringy” Chern classes for singular varieties. These classes enjoy strong birational invariance properties, and their definition encodes data coming from resolution of singularities. The singularities allowed in the theory are those typical of the minimal model program; examples are given by quotients of manifolds by finite groups. For the latter an explicit formula is proven, assuming that the canonical line bundle of the manifold descends to the quotient. This gives an expression of the stringy Chern class of the quotient in terms of Chern–Schwartz–MacPherson classes of the fixed-point set data.

Metrics of positive Ricci curvature on quotient spaces

One of the classical problems in differential geometry is the investigation of closed manifolds which admit Riemannian metrics with given lower bounds for the sectional or the Ricci curvature and the study of relations between the existence of such metrics and the topology and geometry of the underlying manifold. Despite many efforts during the past decades, this problem is still far from being understood. For example, so far the only obstructions to the existence of a metric with positive Ricci curvature come from the obstructions to the existence of metrics with positive scalar curvature ([Li], [Hi], [Ro], [SchY], [Ta]) and the Bonnet-Myers theorem which implies that the fundamental group of a closed manifold with positive Ricci curvature must be finite. Fruitful constructions of metrics with positive Ricci curvature on closed manifolds have so far been established by techniques that include deformation of metrics ([Au], [Eh], [We]), Kahler geometry ([Yau1], [Yau2]), bundles and warping ([Po], [Na], [BB], [GPT]), special kinds of surgery ([SY], [Wr]), metrical glueing ([GZ2]), and Sasakian geometry ([BGN]). Particularly large classes of examples of manifolds with positive Ricci curvature are given by all compact homogeneous spaces with finite fundamental group ([Na]) and all closed cohomogeneity one manifolds with finite fundamental group ([GZ2]). In this article we present several new classes of manifolds which admit metrics of positive or nonnegative Ricci curvature. The idea is to consider quotients of manifolds (M,g) of positive or nonnegative Ricci curvature by a free isometric action. While taking such a quotient non-decreases the sectional curvature, it may well happen in general that the Ricci curvature of the quotient is inferior to the one of M . Our first results, however, state that the quotient of M does admit metrics of positive Ricci curvature if M belongs to one of the aforementioned classes of homogeneous spaces or spaces with a cohomogeneity one action. More precisely, we prove:

Spaces on and beyond the boundary of existence

In this note we discuss various questions on whether or not quotients of Riemannian manifolds by Lie groups can be the Gromov-Hausdorff limits of manifolds with certain curvature bounds. In particular we show that any quotient of a manifold by a Lie group is a limit of manifolds with a lower curvature bound; this answers a question posed by Burago, Gromov, and Perelman. On the other hand, we prove that not all such spaces are limits of manifolds with absolute curvature bounds. We also give examples of spaces with curvature ≥1 that are not limits of manifolds with curvature ≥δ > 1/4.

On quotients of manifolds: A generalization of the closed subgroup theorem

On quotients of manifolds: A generalization of the closed subgroup theorem