Discretization of harmonic measures for foliated bundles

We prove in this Note that there is, for some foliated bundles, a bijective correspondence between Garnett?s harmonic measures and measures on the fiber that are stationary for some probability measure on the holonomy group. As a consequence, we show the uniqueness of the harmonic measure in the case of some foliations transverse to projective fiber bundles.     

Liouvillian integration and Bernoulli foliations

Analytic foliations in the 2-dimensional complex projective space with algebraic invariant curves are studied when the holonomy groups of these curves are solvable. It is shown that such a condition leads to the existence of a Liouville type first integral, and, under ``generic' extra conditions, it is proven that these foliations can be defined by Bernoulli equations.

     

Projective holonomy I: principles and properties

The aim of this paper and its sequel is to introduce and classify the holonomy algebras of the projective Tractor connection. After a brief historical background, this paper presents and analyses the projective Cartan and Tractor connections, the various structures they can preserve, and their geometric interpretations. Preserved subbundles of the Tractor bundle generate foliations with Ricci-flat leaves. Contact- and Einstein-structures arise from other reductions of the Tractor holonomy, as do U(1) and bundles over a manifold of smaller dimension.     

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.     

On codimension one foliations with prescribed cuspidal separatrix

In this paper, we will construct a pre-normal form for germs of codimension one holomorphic foliation having a particular type of separatrix, of cuspidal type. As an application, we will explain how this form could be used in order to study the analytic classification of the singularities via the projective holonomy, in the generalized surface case. We will also give an application to the analytic classification of singularities, and a sufficient condition, in the quasi-homogeneous, three-dimensional case, for these foliations to be of generalized surface type.     

Growth of leaves in transversely affine foliations

In this note we give estimates for the growth of leaves in transversely affine foliations which depend on the properties of the affine holonomy group.

     

Equicontinuous foliated spaces

Riemannian foliations are characterized as those foliations whose holonomy pseudogroup consists of local isometries of a Riemannian manifold. Their main structural features are well understood since the work of Molina. In this paper we analyze the more general concept of equicontinuous pseudogroup of homeomorphisms, which gives rise to the notion of equicontinuous foliated space. We show that equicontinuous foliated spaces have structural properties similar to those known for Riemannian foliations: the universal covers of their leaves are in the same quasi-isometry class, leaf closures are homogeneous spaces, and the holonomy pseudogroup is indeed given by local isometries.     

Parallel submanifolds of complex projective space and their normal holonomy

The object of this article is to compute the holonomy group of the normal connection of complex parallel submanifolds of the complex projective space. We also give a new proof of the classification of complex parallel submanifolds by using a normal holonomy approach. Indeed, we explain how these submanifolds can be regarded as the unique complex orbits of the (projectivized) isotropy representation of an irreducible Hermitian symmetric space. Moreover, we show how these important submanifolds are related to other areas of mathematics and theoretical physics. Finally, we state a conjecture about the normal holonomy group of a complete and full complex submanifold of the complex projective space. Research partially supported by GNSAGA (INdAM) and MIUR of Italy.     

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.     

Foliations

The survey is based on works on the theory of foliations reviewed in RZhMatematika during 1970–1979. The basic topics are the classification of foliations, characteristic classes, the qualitative theory of foliations (holonomy, growth of leaves, etc.), and special classes of foliations (compact foliations, Riemannian foliations, etc.).Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 18, pp. 151–213, 1981.     

A Global Stability Theorem for Transversely Holomorphic Foliations

We prove a global stability theorem for transversely holomorphic foliations of complex codimension one: if there exists a compact leaf with finite holonomy, then the foliation is a Seifert fibration (that is, every leaf is compact and has finite holonomy).     

On degeneracy schemes of maps of vector bundles and applications to holomorphic foliations

In this paper we provide sufficient conditions for maps of vector bundles on smooth projective varieties to be uniquely determined by their degeneracy schemes. We then specialize to holomorphic distributions and foliations. In particular, we provide sufficient conditions for foliations of arbitrary rank on $\mathbb P ^n$ to be uniquely determined by their singular schemes.     

Projective holonomy II: cones and complete classifications

The aim of this paper and its prequel is to introduce and classify the irreducible holonomy algebras of the projective Tractor connection. This is achieved through the construction of a ‘projective cone’, a Ricci-flat manifold one dimension higher whose affine holonomy is equal to the Tractor holonomy of the underlying manifold. This paper uses the result to enable the construction of manifolds with each possible holonomy algebra.     

Leaf closures of Riemannian foliations: A survey on topological and geometric aspects of Killing foliations

A smooth foliation is Riemannian when its leaves are locally equidistant. The closures of the leaves of a Riemannian foliation on a simply-connected manifold, or more generally of a Killing foliation, are described by flows of transverse Killing vector fields. This offers significant technical advantages in the study of this class of foliations, which nonetheless includes other important classes, such as those given by the orbits of isometric Lie group actions. Aiming at a broad audience, in this survey we introduce Killing foliations from the very basics, starting with a brief revision of the main objects appearing in this theory, such as pseudogroups, sheaves, holonomy and basic cohomology. We then review Molino’s structural theory for Riemannian foliations and present its transverse counterpart in the theory of complete pseudogroups of isometries, emphasizing the connections between these topics. We also survey some classical results and recent developments in the theory of Killing foliations. Finally, we review some topics in the theory of singular Riemannian foliations, including the recent proof of Molino’s conjecture, and discuss singular Killing foliations.     

Difféomorphismes holomorphes Anosov

We classify the holomorphic diffeomorphisms of complex projective varieties with an Anosov dynamics and holomorphic stable and unstable foliations: The variety is finitely covered by a compact complex torus and the diffeomorphism corresponds to a linear transformation of this torus.

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Difféomorphismes holomorphes Anosov

     

Tangential category of foliations

We present techniques to construct tangential homotopies of subsets of foliated manifolds and use these to obtain bounds and explicit computations for the tangential Lusternik-Schnirelmann category of foliations. For example, we show that this number is not greater than the dimension of the foliation, that it is an upper semi-continuous function on the space of p-dimensional foliations of a given manifold, and that it is equal to the dimension of the foliation for all codimension 1 foliations without holonomy on compact nilmanifolds.     

On the finsler space admitting a holonomy group

Summary The ideas of holonomy group fixing an m-dimensional plane in a Finsler space were given by one of the present authors[1]. In that paper the deformation properties of the space admitting such holonomy group were of main consideration and, indeed, the decomposition characteristics of the space were not touched upon. In the present paper we consider the decomposition of the space due to the existence of holonomy group. The geometry is constructed on the decomposed metric of the space. The decomposition properties of various entities such as the connection parameters, the covariant derivatives, the curvature tensors, and the projective curvature tensors have been studied. In all there are six articles in the paper. The first of these is introductory. The next three articles are dealt with the Cartan's approach to Finsler space whereas the fifth one is dealt with Berwald's approach. The last article is devoted to the theory of decomposition in the projective curvature tensors. Entrata in Redazione il 18 ottobre 1969.     

Local structure of Koszul-Vinberg and of Lie algebroids

In this short note we continue our study of Koszul-Vinberg algebroids which form a subcategory of the category of Lie algebroids, and which appear naturally in the study of affine structures, affine and transversally affine foliations [N. Nguiffo Boyom, R. Wolak, J. Geom. Phys. 42 (2002) 307-317]. We prove a local decomposition theorem for KV-algebroids. Using the notion of KV-algebroids we introduce a new class of singular foliations: affine singular foliations. In the last section we study the holonomy of these foliations and prove a stability theorem.     

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.