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.

The following theorem, which includes as very special cases results of Jouanolou and Hrushovski on algebraic $D$-varieties on the one hand, and of Cantat on rational dynamics on the other, is established: Working over a field of characteristic zero, suppose $\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}:Z\rightarrow X$ are dominant rational maps from an (possibly nonreduced) irreducible scheme $Z$ of finite type to an algebraic variety $X$, with the property that there are infinitely many hypersurfaces on $X$ whose scheme-theoretic inverse images under $\unicode[STIX]{x1D719}_{1}$ and $\unicode[STIX]{x1D719}_{2}$ agree. Then there is a nonconstant rational function $g$ on $X$ such that $g\unicode[STIX]{x1D719}_{1}=g\unicode[STIX]{x1D719}_{2}$. In the case where $Z$ is also reduced, the scheme-theoretic inverse image can be replaced by the proper transform. A partial result is obtained in positive characteristic. Applications include an extension of the Jouanolou–Hrushovski theorem to generalised algebraic ${\mathcal{D}}$-varieties and of Cantat’s theorem to self-correspondences.

We construct an example of a fixed point free Anosov diffeomorphism of the plane, which is not topologically conjugate to a translation.

We prove the analogue of the Ax–Lindemann–Weierstrass theorem for not necessarily arithmetic lattices of the automorphism group of the complex unit ball $\mathbb{B}^{n}$ using methods of several complex variables, algebraic geometry and Kähler geometry. Consider a torsion-free lattice $\unicode[STIX]{x1D6E4}\,\subset \,\text{Aut}(\mathbb{B}^{n})$ and the associated uniformization map $\unicode[STIX]{x1D70B}:\mathbb{B}^{n}\rightarrow \mathbb{B}^{n}/\unicode[STIX]{x1D6E4}=:X_{\unicode[STIX]{x1D6E4}}$. Given an algebraic subset $S\,\subset \,\mathbb{B}^{n}$ and writing $Z$ for the Zariski closure of $\unicode[STIX]{x1D70B}(S)$ in $X_{\unicode[STIX]{x1D6E4}}$ (which is equipped with a canonical quasi-projective structure), in some precise sense we realize $Z$ as a variety uniruled by images of algebraic subsets under the uniformization map, and study the asymptotic geometry of an irreducible component $\widetilde{Z}$ of $\unicode[STIX]{x1D70B}^{-1}(Z)$ as $\widetilde{Z}$ exits the boundary $\unicode[STIX]{x2202}\mathbb{B}^{n}$ by exploiting the strict pseudoconvexity of $\mathbb{B}^{n}$, culminating in the proof that $\widetilde{Z}\,\subset \,\mathbb{B}^{n}$ is totally geodesic. Our methodology sets the stage for tackling problems in functional transcendence theory for arbitrary lattices of $\text{ Aut}(\unicode[STIX]{x1D6FA})$ for (possibly reducible) bounded symmetric domains $\unicode[STIX]{x1D6FA}$.

We prove that a leafwise homotopy equivalence between compact foliated manifolds induces a well defined bounded operator between all Sobolov spaces of leafwise (for the natural foliations of the graphs of the original foliations) differential forms with coefficients in a leafwise flat bundle. We further prove that the associated map on the leafwise reduced L2 cohomology is an isomorphism which only depends on the leafwise homotopy class of the homotopy equivalence.

We study one-forms with zero wedge-product, which we call collinear, and their foliations. We characterise the set of forms that define a given foliation, with special attention to closed forms and forms with small singular sets. We apply the notion of collinearity to give a criterion for the existence of a compact leaf and to study homological properties of compact leaves.

We investigate in this paper the topological stability of pairs (ω, X), where ω is a germ of an integrable 1-form and X is a germ of a vector field tangent to the foliation determined by ω.

We review basic notions and methods of noncommutative geometry and their applications to analysis and geometry on foliated manifolds.

When the index bundle of a longitudinal Dirac type operator is transversely smooth, we define its Chern character in Haefliger cohomology and relate it to the Chern character of the K—theory index. This result gives a concrete connection between the topology of the foliation and the longitudinal index formula. Moreover, the usual spectral assumption on the Novikov-Shubin invariants of the operator is improved.

An analogue of the Stein embedding theorem for $C^\infty$ manifolds endowed with two equidimensional supplementary foliations is proved.

AMS 2000 Mathematics subject classification: Primary 30G35. Secondary 16P10; 26E05; 32E10; 53C12; 57R30.

For any Riemannian foliation $\cal F$ on a closed manifold M with an arbitrary bundle-like metric, leafwise heat flow of differential forms is proved to preserve smoothness on M at infinite time. This result and its proof have consequences about the space of bundle-like metrics on M, about the dimension of the space of leafwise harmonic forms, and mainly about the second term of the differentiable spectral sequence of $\cal F$.

For foliations on Riemannian manifolds, we develop elementary geometric and topological properties of the mean curvature one-form κ and the normal plane field one-form β. Through examples, we show that an important result of Kamber-Tondeur on κ is in general a best possible result. But we demonstrate that their bundle-like hypothesis can be relaxed somewhat in codimension 2. We study the structure of umbilic foliations in this more general context and in our final section establish some analogous results for flows.

Using a combination of optical microscopy and X-ray texture goniometry, an integrated microstructural and crystallographic fabric study has been made of quartz mylonites from thrust sheets located beneath, but immediately adjacent to, the Moine thrust in the Assynt and Eriboll regions of NW Scotland. A correlation is established between shape fabric symmetry and pattern of crystallographic preferred orientation, a particularly clear relationship being observed between shape fabric variation and quartz a-axis fabrics.

Coaxial strain paths dominate the internal parts of the thrust sheets and are indicated by quartz c- and a-axis fabrics which are symmetrical with respect to foliation and lineation. Non-coaxial strain paths are indicated within the more intensely deformed quartzites located near the boundaries of the sheets by asymmetrical c- and a-axis fabrics. These kinematic interpretations are supported by microstructural studies. At the Stack of Glencoul in the northern part of the Assynt region, the transition zone between these kinematic (strain path) domains is located at approximately 20 cm beneath the Moine thrust and is marked by a progression from symmetrical cross-girdle c-axis fabrics (30cm beneath the thrust), through asymmetrical cross-girdle c-axis fabrics to asymmetrical single girdle c-axis fabrics (0·5 cm beneath the thrust).

Tectonic models (incorporating processes such as extensional flow, gravity spreading and tectonic loading) which may account for the presence of strain path domains within the thrust sheets are considered, and their compatibility with local thrust sheet geometries assessed.

Graphite bodies of economic importance occur in the Eastern Ghats migmatite complex of Orissa, India, which consists of sillimanite-rich gneiss (khondalite), calc-silicate granulite, basic (pyroxene) granulite and acidic gneiss palaeosomes and a variety of quartzofeldspathic neosomes and pegmatite veins. There is more graphite in migmatised khondalite and calc-silicate granulite than in their unmigmatised counterparts and it is common at pegmatite-country rock margins. Lodes of graphite are structurally controlled by fold hinge zones, necks of boudins and along the dominant foliation in rocks that show the effects of polyphase deformation. Faults, fractures and joints also act as structural controls. The richest lodes are located where there is the coincidence of several lithological and structural features.

The graphite occurs in graphite-rich schists and gneisses, as veins and as disseminated flakes along grain boundaries and in microfractures and cleavages. It is present in all rock types except pegmatites emplaced after the D4 deformational phase, and is localised in structures developed during the D1, D2, D3 and D4 phases. Most of the bodies of graphite are lenticular and show pinch and swell characters. Fourteen textural and microstructural habits have been recognised and are described with many of the features of fabric, such as intergrowths, inclusions and kinks being particularly well shown under reflected light.

Methanation of carbonate minerals is considered the most likely initial stage in the formation of the carbon with subsequent pyrolysis of the methane being the process of graphitisation. While the possibility of the carbon being either of sedimentary derivation orfrom igneous-derived CO2 cannot be ruled out, there is no convincing evidence for such sources. Remobilisation of graphite following its formation by pyrolysis of methane in structurally controlled low pressure zones is the dominant mechanism of concentration. Thefabric features point to the graphite having moved as a solid phase along open spaces inthe gangue minerals while still maintaining its crystallinity, and having been deformed in the process. The effects of remobilisation apparently have obliterated evidence relating to other methods of concentration.

Fourteen granitoid plutons studied in the Baie d'Espoir area of southern Newfoundland are divisible into a northern group of eight and a southern group of six. The former tend to be undeformed, have thermal aureoles, and intrude greenschist facies Ordovician and Silurian rocks of the Dunnage zone following a period of folding and thrusting towards the SE. Four Rb-Sr whole-rock isochrons yield similar ages around 430 Ma for these plutons, fixing the deformation within early Silurian times. The latter plutons syntectonically intruded without aureole effects, and are all foliated and elongate parallel to the regional fabric of amphibolite facies gneisses and migmatites of the Gander zone. Whole-rock Rb-Sr isochrons around 350 Ma for both syntectonic and post-tectonic plutons indicate a rapid metamorphism, deformation, intrusion and uplift of the Gander zone in this area. There appear to be no systematic relationships between petrological and geochemical characteristics of the plutons and their tectono-stratigraphic setting. This study conclusively invalidates earlier hypotheses that deformed granitoid rocks of the Gander zone are older than the undeformed granitoids of the Dunnage zone; indeed, just the reverse is true.

The massive pyrite-pyrrhotite-chalcopyrite-sphalerite deposit of Outokumpu, comprising the Keretti and Vuonos orebodies, is a deformed and metamorphosed strata-bound mass associated with mineralised stockworks. Mobilisation of much of the ore followed formation of large recumbent isoclinal folds that are the major structures of the surrounding rocks and associated with the modification of originally flat saucer-shaped ore lenses into elongate ruler-shaped masses. Further modification of shape took place at the mobilisation stage with much of the pyrrhotitic ore, particularly, now occupying the thickest parts of the orebodies in the form of breccia or microbreccia. In many parts gross original characters still exist and the pyritic and pyrrhotitic constituents of the ore have survived as separate entities while locally the pyritic ore retains pre-deformational characteristics and consistent stratigraphic position within a thin horizon.

Both ore and country rocks show evidence of extensive polyphase deformation with the effects of six fold phases shown in the ore. Mineral assemblages in the country rocks indicate a middle amphibolite facies peak of metamorphism. The serpentinite-black schist-carbonate-quartzite rock assemblage, with which the ore is associated, was tectonically incorporated within the regionally extensive mica schist by even earlier subhorizontal thrusting. This is related to the movement of a thrust nappe with the interdigitation of an ocean-floor ophiolite assemblage and flysch deposited during ocean closure associated with Svecokarelian tectonism.

The original formation of the Keretti and Vuonos sulphide masses took place in a marine exhalative environment with a pyritic layer overlying a pyrrhotitic layer in each of the two c. 4 km diameter irregularly oval-shaped depressions whose centres were c. 8 km apart. The mineralised stockwork below each mass represents the upper parts of the conduit for metalbearing fluids in a convective system.