On the existence of stable compact leaves for transversely holomorphic foliations

We prove that a transversely holomorphic foliation on a compact manifold exhibits some compact leaf with finite holonomy group, provided that the set of compact leaves is not a zero measure set. A similar result is stated for groups of complex diffeomorphisms and periodic orbits.     

The first cohomology group of leaves and local stability of compact foliations

A foliation with all leaves compact (compact foliation) is called locally stable if every leaf has a basis of neighborhoods which are unions of leaves. We study the relationship between the first real cohomology group of leaves and the local stability of compact foliations. We show by example that the topology of the typical leaves (i.e. leaves with zero holonomy) has no influence on the local stability of the foliation while — at least for small codimensions — (less than 4 in general or less than 5 for foliations on compact minifolds) — a locally unstable foliation has a leaf F with infinite holonomy and a finite covering F' of F such that H¹(F'; IR) O. We also prove a related structural stability result for fibre bundles.     

Curves of zero self-intersection and foliations

We study the holonomy group of a holomorphic foliation in a surface along a compact leaf. It is shown that linearization of a neighborhood of the curve implies strong restrictions in those groups.     

Bounded geometry and leaves

The main theorem states that any complete connected Riemannian manifold of bounded geometry can be isometrically realized as a leaf with trivial holonomy in a compact Riemannian foliated space.     

Smoothness of holonomy covers for singular foliations and essential isotropy

Given a singular foliation, we attach an “essential isotropy” group to each of its leaves, and show that its discreteness is the integrability obstruction of a natural Lie algebroid over the leaf. We show that a condition ensuring discreteness is the local surjectivity of a transversal exponential map associated with the maximal ideal of vector fields prescribed to be tangent to the foliation. The essential isotropy group is also shown to control the smoothness of the holonomy cover of the leaf (the associated fiber of the holonomy groupoid), as well as the smoothness of the associated isotropy group. Namely, the (topological) closeness of the essential isotropy group is a necessary and sufficient condition for the holonomy cover to be a smooth (finite-dimensional) manifold and the isotropy group to be a Lie group. These results are useful towards understanding the normal form of a singular foliation around a compact leaf. At the end of this article we briefly outline work of ours on this normal form, to be presented in a subsequent paper.     

Singular Riemannian foliations on simply connected spaces

A singular foliation on a complete Riemannian manifold is said to be Riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. The singular foliation is said to admit sections if each regular point is contained in a totally geodesic complete immersed submanifold that meets every leaf orthogonally and whose dimension is the codimension of the regular leaves. A typical example of such a singular foliation is the partition by orbits of a polar action, e.g. the orbits of the adjoint action of a compact Lie group on itself.We prove that a singular Riemannian foliation with compact leaves that admits sections on a simply connected space has no exceptional leaves, i.e., each regular leaf has trivial normal holonomy. We also prove that there exists a convex fundamental domain in each section of the foliation and in particular that the space of leaves is a convex Coxeter orbifold.     

Parallel transports associated to stochastic holonomies

A stochastic holonomy along a loop obtained from the OU process on the path space over a compact Riemannian manifold is computed. The result shows that the stochastic holonomy just gives the parallel transport with respect to the Markov connection along the OU process on the path space     

Twistor theory and Blaschke manifolds

Using the twistor theory on quaternionic Kaehler manifolds and some recent results on Blaschke manifolds and compact manifolds whose holonomy group is Spin (7), we prove that a Blaschke manifold of nonnegative scalar curvature whose holonomy group is exceptional is isometric to a projective space.     

Holonomy transformations for singular foliations

In order to understand the linearization problem around a leaf of a singular foliation, we extend the familiar holonomy map from the case of regular foliations to the case of singular foliations. To this aim we introduce the notion of holonomy transformation. Unlike the regular case, holonomy transformations cannot be attached to classes of paths in the foliation, but rather to elements of the holonomy groupoid of the singular foliation.     

Foliations of codimension 2 with all leaves compact

We show that the following question, due to Haefliger can be answered positively for C¹-foliations of codimenslon 2: if is a foliation on a compact manifold with all leaves compact, does every neighborhood of any leaf F of contain a neighborhood of F which is a union of leaves? In the course of the proof we show that the Euler number of leaves which do not have the above property is zero for an open and dense subset of these leaves if the holonomy groups of these leaves are cyclic. The answer to Haefliger's question in codimension 2 was first and independently obtained by Edwards-Millett-Sullivan [1]     

Longitudinal smoothness of the holonomy groupoid

Iakovos Androulidakis and Georges Skandalis have defined a holonomy groupoid for any singular foliation. This groupoid, whose topology is usually quite bad, is the starting point for the study of longitudinal pseudodifferential calculus on such foliation and its associated index theory. These studies can be highly simplified under the assumption of the holonomy groupoid being longitudinally smooth. In this note, we rephrase the period bounding lemma that asserts that a vector field on a compact manifold admits a strictly positive lower bound for its periodic orbits in order to prove that the holonomy groupoid is always longitudinally smooth.     

A x -operator on complete riemannian manifolds

In this paper we give a generalisation of Kostant’s Theorem about theA _x -operator associated to a Killing vector fieldX on a compact Riemannian manifold. Kostant proved (see [6], [5] or [7]) that in a compact Riemannian manifold, the (1, 1) skew-symmetric operatorA _x =L _x −≡ _x associated to a Killing vector fieldX lies in the holonomy algebra at each point. We prove that in a complete non-compact Riemannian manifold (M, g) theA _x -operator associated to a Killing vector field, with finite global norm, lies in the holonomy algebra at each point. Finally we give examples of Killing vector fields with infinite global norms on non-flat manifolds such thatA _x does not lie in the holonomy algebra at any point.     

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.     

Affine embeddings of flat compact manifolds I

We show that a compact flat manifold has the structure of a real affine variety. Using generators of the affine algebra we constructed embeddings with flat induced metric of all compact flat manifolds with cyclic holonomy group of order 2.     

Affine manifolds with diagonal holonomy

As first defined by Smillie, an affine manifold with diagonal holonomy is a manifold equipped with an atlas such that the changes of charts are restrictions of elements of the subgroup of Aff ( \mathbbRⁿ{\mathbb{R}^n}) formed by diagonal matrices. Refining Smillie’s theorem, Benoist proved that if a compact manifold M is split into manifolds with corners corresponding to complete simplicial fans of a fixed frame by its hypersurfaces with normal crossing, then the product of M and a torus of suitable dimension is a finite cover of an affine manifold with diagonal holonomy, and conversely. Motivated by the result of Benoist, we introduce a “Benoist manifold” and a natural definition of complexity for them. In particular, we study some properties of “Benoist manifolds”.     

Spin Structures on Flat Manifolds

The aim of this paper is to present some results about spin structures on flat manifolds. We prove that any finite group can be the holonomy group of a flat spin manifold. Moreover, we shall give some methods of constructing spin structures related to the holonomy representation.     

Spin Structures on Flat Manifolds

The aim of this paper is to present some results about spin structures on flat manifolds. We prove that any finite group can be the holonomy group of a flat spin manifold. Moreover, we shall give some methods of constructing spin structures related to the holonomy representation.     

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.     

Equifocality of a singular Riemannian foliation

A singular foliation on a complete Riemannian manifold is said to be Riemannian if each geodesic that is perpendicular to a leaf at one point remains perpendicular to every leaf it meets. We prove that the regular leaves are equifocal, i.e., the end point map of a normal foliated vector field has constant rank. This implies that we can reconstruct the singular foliation by taking all parallel submanifolds of a regular leaf with trivial holonomy. In addition, the end point map of a normal foliated vector field on a leaf with trivial holonomy is a covering map. These results generalize previous results of the authors on singular Riemannian foliations with sections.

     

Stable ergodicity and julienne quasi-conformality

In this paper we dramatically expand the domain of known stably ergodic, partially hyperbolic dynamical systems. For example, all partially hyperbolic affine diffeomorphisms of compact homogeneous spaces which have the accessibility property are stably ergodic. Our main tools are the new concepts – julienne density point and julienne quasi-conformality of the stable and unstable holonomy maps. Julienne quasi-conformal holonomy maps preserve all julienne density points. Received June 14, 1999 / final version received October 25, 1999