Quasigeodesic behavior of flow lines is a very useful property in the study of Anosov flows. Not every Anosov flow in dimension three is quasigeodesic. In fact, until recently, up to orbit equivalence, the only previously known examples of quasigeodesic Anosov flows were suspension flows. In a recent article, the second author proved that an Anosov flow on a hyperbolic 3-manifold is quasigeodesic if and only if it is non-$\mathbb {R}$-covered, and this result completes the classification of quasigeodesic Anosov flows on hyperbolic 3-manifolds. In this article, we prove that a new class of examples of Anosov flows are quasigeodesic. These are the first examples of quasigeodesic Anosov flows on 3-manifolds that are neither Seifert, nor solvable, nor hyperbolic. In general, it is very hard to show that a given flow is quasigeodesic and, in this article, we provide a new method to prove that an Anosov flow is quasigeodesic.

The Haefliger–Thurston conjecture predicts that Haefliger's classifying space for $C^r$-foliations of codimension $n$ whose normal bundles are trivial is $2n$-connected. In this paper, we confirm this conjecture for piecewise linear (PL) foliations of codimension $2$. Using this, we use a version of the Mather–Thurston theorem for PL homeomorphisms due to the author to derive new homological properties for PL surface homeomorphisms. In particular, we answer the question of Epstein in dimension $2$ and prove the simplicity of the identity component of PL surface homeomorphisms.

We show that if an open set in $\mathbb{R}^d$ can be fibered by unit n-spheres, then $d \geq 2n+1$, and if $d = 2n+1$, then the spheres must be pairwise linked, and $n \in \left\{0, 1, 3, 7 \right\}$. For these values of n, we construct unit n-sphere fibrations in $\mathbb{R}^{2n+1}$.

We construct one-dimensional foliations which are subfoliations of two-dimensional foliations in $3$ -manifolds. The subfoliation is by quasigeodesics in each two-dimensional leaf, but it is not funnel: not all quasigeodesics share a common ideal point in most leaves.

Following Losik’s approach to Gelfand’s formal geometry, certain characteristic classes for codimension-one foliations coming from the Gelfand-Fuchs cohomology are considered. Sufficient conditions for nontriviality in terms of dynamical properties of generators of the holonomy groups are found. The nontriviality for the Reeb foliations is shown; this is in contrast with some classical theorems on the Godbillon-Vey class; for example, the Mizutani-Morita-Tsuboi theorem about triviality of the Godbillon-Vey class of foliations almost without holonomy is not true for the classes under consideration. It is shown that the considered classes are trivial for a large class of foliations without holonomy. The question of triviality is related to ergodic theory of dynamical systems on the circle and to the problem of smooth conjugacy of local diffeomorphisms. Certain classes are obstructions for the existence of transverse affine and projective connections.

Using open books, we prove the existence of a non-vanishing steady solution to the Euler equations for some metric in every homotopy class of non-vanishing vector fields of any odd-dimensional manifold. As a corollary, any such field can be realized in an invariant submanifold of a contact Reeb field on a sphere of high dimension. The solutions constructed are geodesible and hence of Beltrami type, and can be modified to obtain chaotic fluids. We characterize Beltrami fields in odd dimensions and show that there always exist volume-preserving Beltrami fields which are neither geodesible nor Euler flows for any metric. This contrasts with the three-dimensional case, where every volume-preserving Beltrami field is a steady Euler flow for some metric. Finally, we construct a non-vanishing Beltrami field (which is not necessarily volume-preserving) without periodic orbits in every manifold of odd dimension greater than three.

We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint, 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen–Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a 3-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan’s homological characterization of geodesible flows in the volume-preserving case. As an application, we show that steady Euler flows cannot be constructed using plugs (as in Wilson’s or Kuperberg’s constructions). Analogous results in higher dimensions are also proved.

We extend the unpublished work of Handel and Miller on the classification, up to isotopy, of endperiodic automorphisms of surfaces. We give the Handel–Miller construction of the geodesic laminations, give an axiomatic theory for pseudo-geodesic laminations, show that the geodesic laminations satisfy the axioms, and prove that pseudo-geodesic laminations satisfying our axioms are ambiently isotopic to the geodesic laminations. The axiomatic approach allows us to show that the given endperiodic automorphism is isotopic to a smooth endperiodic automorphism preserving smooth laminations ambiently isotopic to the original ones. Using the axioms, we also prove the ‘transfer theorem’ for foliations of 3-manifolds, namely that, if two depth-one foliations ${\mathcal{F}}$ and ${\mathcal{F}}^{\prime }$ are transverse to a common one-dimensional foliation ${\mathcal{L}}$ whose monodromy on the non-compact leaves of ${\mathcal{F}}$ exhibits the nice dynamics of Handel–Miller theory, then ${\mathcal{L}}$ also induces monodromy on the non-compact leaves of ${\mathcal{F}}^{\prime }$ exhibiting the same nice dynamics. Our theory also applies to surfaces with infinitely many ends.

We prove that a class of weakly partially hyperbolic endomorphisms on $\mathbb{T}^{2}$ are dynamically coherent and leaf conjugate to linear toral endomorphisms. Moreover, we give an example of a partially hyperbolic endomorphism on $\mathbb{T}^{2}$ which does not admit a centre foliation.

Let $f:M\rightarrow M$ be a dynamically coherent partially hyperbolic diffeomorphism whose center foliation has all its leaves compact. We prove that if the unstable bundle of $f$ is one-dimensional, then the volume of center leaves must be bounded in $M$.

A classic result due to Furstenberg is the strict ergodicity of the horocycle flow for a compact hyperbolic surface. Strict ergodicity is unique ergodicity with respect to a measure of full support, and therefore it implies minimality. The horocycle flow has been previously studied on minimal foliations by hyperbolic surfaces on closed manifolds, where it is known not to be minimal in general. In this paper, we prove that for the special case of Riemannian foliations, strict ergodicity of the horocycle flow still holds. This, in particular, proves that this flow is minimal, which establishes a conjecture proposed by Matsumoto. The main tool is a theorem due to Coudène, which he presented as an alternative proof for the surface case. It applies to two continuous flows defining a measure-preserving action of the affine group of the line on a compact metric space, precisely matching the foliated setting. In addition, we briefly discuss the application of Coudène’s theorem to other kinds of foliations.

We develop the transversal harmonic theory for a transversally symplectic flow on a manifold and establish the transversal hard Lefschetz theorem. Our main results extend the cases for a contact manifold (H. Kitahara and H. K. Pak, ‘A note on harmonic forms on a compact manifold’, Kyungpook Math. J.43 (2003), 1–10) and for an almost cosymplectic manifold (R. Ibanez, ‘Harmonic cohomology classes of almost cosymplectic manifolds’, Michigan Math. J.44 (1997), 183–199). For the point foliation these are the results obtained by Brylinski (‘A differential complex for Poisson manifold’, J. Differential Geom.28 (1988), 93–114), Haller (‘Harmonic cohomology of symplectic manifolds’, Adv. Math.180 (2003), 87–103), Mathieu (‘Harmonic cohomology classes of symplectic manifolds’, Comment. Math. Helv.70 (1995), 1–9) and Yan (‘Hodge structure on symplectic manifolds’, Adv. Math.120 (1996), 143–154).

The fundamental germ is a generalization of π1, first defined for laminations which arise through group actions [4]. In this paper, the fundamental germ is extended to any lamination having a dense leaf admitting a smooth structure. In addition, an amplification of the fundamental germ called the mother germ is constructed, which is, unlike the fundamental germ, a topological invariant. The fundamental germs of the antenna lamination and the PSL(2,ℤ) lamination are calculated, laminations for which the definition in [4] was not available. The mother germ is used to give a new proof of a Nielsen theorem for the algebraic universal cover of a closed surface of hyperbolic type.

Nous démontrons des théorèmes d'immersion holomorphe dans un espace projectif complexe pour des variétés kählériennes non compactes et des laminations par variétés complexes qui admettent un fibré en droites holomorphe strictement positif. En particulier, nous montrons que sur une lamination compacte par surfaces de Riemann, les fonctions méromorphes séparent les points si et seulement si aucun cycle feuilleté n'est homologue à $0$.

We prove holomorphic immersion theorems in a finite dimensional complex projective space for kählerian non-compact manifolds and for laminations by complex manifolds that carry a line bundle of positive curvature. In particular, we prove that on a Riemann surfaces lamination of a compact space, the space of meromorphic functions separates points if and only if every foliation cycle is non homologous to $0$.

Recently [8], a harmonic theory was developed for a compact contact manifold from the viewpoint of the transversal geometry of contact flow. A contact flow is a typical example of geodesible flow. As a natural generalization of the contact flow, the present paper develops a harmonic theory for various flows on compact manifolds. We introduce the notions of $H$-harmonic and ${{H}^{*}}$-harmonic spaces associated to a Hörmander flow. We also introduce the notions of basic harmonic spaces associated to a weak basic flow. One of our main results is to show that in the special case of isometric flow these harmonic spaces are isomorphic to the cohomology spaces of certain complexes. Moreover, we find an obstruction for a geodesible flow to be isometric.

Given a $p$-dimensional oriented foliation of an $n$-dimensional compact manifold ${{M}^{n}}$ and a transversal invariant measure $\tau$, Sullivan has defined an element of ${{H}_{p}}\left( {{M}^{n}},\,R \right)$. This generalized the notion of a $\mu$-asymptotic cycle, which was originally defined for actions of the real line on compact spaces preserving an invariant measure $\mu$. In this one-dimensional case there was a natural 1–1 correspondence between transversal invariant measures $\tau$ and invariant measures $\mu$ when one had a smooth flow without stationary points.

For what we call an oriented action of a connected Lie group on a compact manifold we again get in this paper such a correspondence, provided we have what we call a positive quantifier. (In the one-dimensional case such a quantifier is provided by the vector field defining the flow.) Sufficient conditions for the existence of such a quantifier are given, together with some applications.

It is shown that a suitable conformai change of the metric in the leaf direction of a transversally oriented Riemannian foliation on a closed manifold will make the basic component of the mean curvature harmonic. As a corollary, we deduce vanishing and finiteness theorems for Riemannian foliations without assuming the harmonicity of the basic mean curvature.

We present first examples of complete transversely affine foliations on compact manifolds with leaves whose closures are not submanifolds. Moreover, we prove that under some additional assumptions the closures of leaves form a singular foliation.

With the general assumption that the manifold admits two orthogonal complementary foliations, one of which is totally geodesic, we study the components of the curvature tensor field of the characteristic connection.

In the case where the manifold is compact, orientable of dimension 6 or 8 and the dimension of the totally geodesic foliation is 4, we relate the sign of the Euler characteristic of the manifold and that of the sectional curvature of the leaves of both foliations.