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In this article we study Stäckel representations of stationary KdV systems. Using Lax formalism we prove that these systems have two different representations as separable Stäckel systems of Benenti type, related with different foliations of the stationary manifold. We do it by constructing an explicit transformation between the jet coordinates of stationary KdV systems and separation variables of the corresponding Benenti systems for arbitrary number of degrees of freedom. Moreover, on the stationary manifold, we present the explicit form of Miura map between both representations of stationary KdV systems, which also yields their bi-Hamiltonian formulation. [ABSTRACT FROM AUTHOR]

We study the problem of classifying local projective structures in dimension two having non trivial Lie symmetries. In particular we obtain a classification of foliated projective structures having positive dimensional Lie algebra of projective vector fields. [ABSTRACT FROM AUTHOR]

In this paper, we discuss how to travel along horizontal broken geodesics of a homogeneous Finsler submersion, i.e., we study, what in Riemannian geometry was called by Wilking, the dual leaves. More precisely, we investigate the attainable sets A q (C) of the set of analytic vector fields C determined by the family of horizontal unit geodesic vector fields C to the fibers F = { ρ − 1 (c) } of a homogeneous analytic Finsler submersion ρ : M → B. Since reverse of geodesics don't need to be geodesics in Finsler geometry, one can have examples on non compact Finsler manifolds M where the attainable sets (the dual leaves) are no longer orbits or even submanifolds. Nevertheless we prove that, when M is compact and the orbits of C are embedded, then the attainable sets coincide with the orbits. Furthermore, if the flag curvature is positive then M coincides with the attainable set of each point. In other words, fixed two points of M , one can travel from one point to the other along horizontal broken geodesics. In addition, we show that each orbit O (q) of C associated to a singular Finsler foliation coincides with M , when the flag curvature is positive, i.e., we prove Wilking's result in Finsler context. In particular we review Wilking's transversal Jacobi fields in Finsler case. [ABSTRACT FROM AUTHOR]

The idea of Lichnerowicz or Morse-Novikov cohomology groups of a manifold has been utilized by many researchers to study important properties and invariants of a manifold. Morse-Novikov cohomology is defined using the differential d ω = d + ω ∧ , where ω is a closed 1-form. We study Morse-Novikov cohomology relative to a foliation on a manifold and its homotopy invariance and then extend it to more general type of forms on a Riemannian foliation. We study the Laplacian and Hodge decompositions for the corresponding differential operators on reduced leafwise Morse-Novikov complexes. In the case of Riemannian foliations, we prove that the reduced leafwise Morse-Novikov cohomology groups satisfy the Hodge theorem and Poincaré duality. The resulting isomorphisms yield a Hodge diamond structure for leafwise Morse-Novikov cohomology. [ABSTRACT FROM AUTHOR]

A stratified space is a kind of topological space together with a partition into smooth manifolds. These kinds of spaces naturally arise in the study of singular algebraic varieties, symplectic reduction, and differentiable stacks. In this paper, we introduce a particular class of stratified spaces called stratified vector bundles, and provide an alternate characterization in terms of monoid actions. We will then provide large families of examples coming from the theory of Whitney stratified spaces, singular foliation theory, and equivariant vector bundle theory. Finally, we extend functorial properties of smooth vector bundles to the stratified case. [ABSTRACT FROM AUTHOR]

We study the twisted Bott-Chern and Aeppli cohomologies on transversely Hermitian foliations. And we study the Frölicher type inequality of these twisted cohomologies. As an application, we prove the Hodge isomorphism on transverse Kähler foliations. [ABSTRACT FROM AUTHOR]

We prove that any holomorphic vector bundle admitting a holomorphic connection, over a compact Kähler Calabi-Yau manifold, also admits a flat holomorphic connection. This addresses a particular case of a question asked by Atiyah and generalizes a result previously obtained in [6] for simply connected compact Kähler Calabi-Yau manifolds. We give some applications of it in the framework of Cartan geometries and foliated Cartan geometries on Kähler Calabi-Yau manifolds. [ABSTRACT FROM AUTHOR]

We describe a procedure, called regularisation, that allows us to study geometric structures on Lie algebroids via foliated geometric structures on manifolds of higher dimension. This procedure applies to various classes of Lie algebroids; namely, those whose singularities are of b k , complex-log, or elliptic type, possibly with self-crossings. One of our main applications is a proof of the Weinstein conjecture for overtwisted b k -contact structures. This was proven in [23] using a certain technical hypothesis. Our approach avoids this assumption by reducing the proof to the foliated setting. As a by-product, we also prove the Weinstein conjecture for other Lie algebroids. [ABSTRACT FROM AUTHOR]

Let M be a manifold endowed with a flat torsionless connection ∇. A symmetric bivector field h on (M , ∇) is said to be a Koszul-Vinberg (K-V for short) bivector field if it satisfies: (∇ α # h) (β , γ) = (∇ β # h) (α , γ) , where α # : = h # (α) and h # : T ⁎ M → T M is the contraction of h. The pair (∇ , h) is called a K-V structure and the triple (M , ∇ , h) is called a K-V manifold. When h is non-degenerate, (M , ∇ , h − 1) is a pseudo-Hessian manifold, otherwise, we have a singular foliation defined by I m h # called the associated affine foliation whose leaves are pseudo-Hessian manifolds. In this paper, we study invariant K-V structures on homogeneous spaces. More precisely, we give an algebraic characterization of these structures in the same spirit of Nomizu's theorem on invariant connections on homogeneous spaces, and we give many classes of examples mainly for reductive or symmetric pairs (G , H). We establish many properties of pseudo-Hessian homogeneous manifolds, in particular, we show that when G is a semi-simple Lie group then G / H does not admit any non trivial G -invariant pseudo-Hessian structure. Finally, we show that the leaves of the affine foliation associated to an invariant K-V structure are homogeneous pseudo-Hessian manifolds. [ABSTRACT FROM AUTHOR]

We consider a 3-dimensional smooth manifold M equipped with an arbitrary, a priori non-integrable, distribution (plane field) D and a vector field T transverse to D. Using a 1-form ω such that D = ker ⁡ ω and ω (T) = 1 we construct a 3-form analogous to that defining the Godbillon-Vey class of a foliation, and show how does this form depend on ω and T. For a compatible Riemannian metric on M , we express this 3-form in terms of the curvature and torsion of normal curves and the non-symmetric second fundamental form of D. We deduce Euler-Lagrange equations of associated functionals: for variable (D , T) on M , and for variable Riemannian metric on (M , D). We show that for a geodesic field T (e.g., for a contact structure) such (D , T) is critical, characterize critical pairs when D is integrable (with examples among twisted products) and prove that these critical pairs are not extrema. [ABSTRACT FROM AUTHOR]

We generalize the notion of fixed point homogeneous isometric group actions to the context of singular Riemannian foliations. We find that in some cases, positively curved manifolds admitting these so-called point leaf maximal SRF's are diffeo/homeomorphic to compact rank one symmetric spaces. In all cases, manifolds admitting such foliations are cohomology CROSSes or finite quotients of them. Among non-simply connected manifolds, we find examples of such foliations which are non-homogeneous. [ABSTRACT FROM AUTHOR]

In this paper, we show that if (T 2 , g) is a class A Lorentzian 2-torus with timelike poles, then there exists a Lipschitz foliation by complete future-directed timelike geodesics with any pre-assigned asymptotic direction in the interior of the stable time cone. This is done by constructing certain C 1 , 1 solutions to the equation g (∇ u , ∇ u) = − 1 on the Abelian cover (R 2 , g). [ABSTRACT FROM AUTHOR]

[Display omitted] Small-scale structures are the primary source of information in interpreting the tectonic evolution of an orogen. However, due to the obliteration and discrete nature of small-scale structures, it is often challenging to ascertain the deformation events. Kink bands are one of the most frequently occurring small-scale structures in foliated rocks and varied tectonic settings, especially in contractional deformational zones. Kink bands in the S-tectonites of Darjeeling-Sikkim Himalaya (DSH) preserve evidence of at least three stages of development of kink bands. The angular parameters of the kink bands (α, β and κ) are used for paleostress reconstruction, estimating volume change and shortening percentages in kink bands developed on the Daling phyllites of the Ramgarh Thrust Sheet in the Tista half-window. Detailed studies on the morphological characteristics of kink bands coupled with structural analysis confirm the formation of kink bands progressively in stages under a hybrid kinking mechanism in a protracted history of deformation. Canonical discriminant analysis of data acquired from microscopic kink bands is employed for erecting a classification scheme with ∼ 90% success rate in predicting the stages of infinitesimal deformation, and validated with mesoscale kink bands in DSH and compared with global examples. The adopted methodology depicts that kink bands can be utilized as a proxy to delineate the extent of volume change and shortening in orogenic belts. [ABSTRACT FROM AUTHOR]

The modular class of a regular foliation is a cohomological obstruction to the existence of a volume form transverse to the leaves which is invariant under the flow of the vector fields of the foliation. It has been used in dynamical systems and operator theory, from which the name modular comes from. By drawing on the relationship between Lie algebroids and regular foliations, this paper extends the notion of modular class to the realm of singular foliations. The singularities are dealt with by replacing the singular foliations by any of their universal Lie ∞-algebroids, and by picking up the modular class of the latter. The geometric meaning of the modular class of a singular foliation F is not as transparent as for regular foliations: it is an obstruction to the existence of a universal Lie ∞-algebroid of F whose Berezinian line bundle is a trivial F -module. This paper illustrates the relevance of using universal Lie ∞-algebroids to extend mathematical notions from regular foliations to singular ones, thus paving the road to defining other characteristic classes in this way. [ABSTRACT FROM AUTHOR]

Abstract Let M be a 2 n -dimensional smooth manifold with a symplectic pair which is a pair of closed 2-forms of constant ranks with complementary kernel foliations. Similar to Moser's stability theorem for symplectic forms, one desires to establish a stability theorem for symplectic pairs. Some sufficient and necessary conditions are obtained by Bande, Ghiggini and Kotschick. In this article, we consider a technical problem relating to the stability theorem. To complete the proof of the stability theorem for symplectic pairs, we verify the smoothness of the isotopy which is ignored in the literature. The Hodge theory for Riemannian foliation is crucial to our discussion. [ABSTRACT FROM AUTHOR]

Abstract We define an L 2 -signature for proper actions on spaces of leaves of transversely oriented foliations with bounded geometry. This is achieved by using the Connes fibration to reduce the problem to the case of Riemannian bifoliations where we show that any transversely elliptic first order operator in an appropriate Beals–Greiner calculus, satisfying the usual axioms, gives rise to a semi-finite spectral triple over the crossed product algebra of the foliation by the action, and hence a periodic cyclic cohomology class through the Connes–Chern character. The Connes–Moscovici hypoelliptic signature operator yields an example of such a triple and gives the differential definition of our " L 2 -signature". For Galois coverings of bounded geometry foliations, we also define an Atiyah–Connes semi-finite spectral triple which generalizes to Riemannian bifoliations the Atiyah approach to the L 2 -index theorem. The compatibility of the two spectral triples with respect to Morita equivalence is proven, and by using an Atiyah-type theorem proven in [7], we deduce some integrality results for Riemannian foliations with torsion-free monodromy groupoids. [ABSTRACT FROM AUTHOR]

Graphical abstract Highlights • Fresh view on the kinematics of a large-scale, long-lived fault in NE Iberia. • New perspectives for analysis of fault zones at shallow crustal levels. • Application of magnetic methods (AMS) to fault analysis. • Comparative analysis from different data sources (structural, magnetic...). Abstract Structural analysis and Anisotropy of Magnetic Susceptibility are here applied to the fault zone (several hundreds of meters thick) associated with an E-W striking, 20 km-long segment of the Vallès-Penedès Fault (Catalan Coastal Ranges). The results obtained indicate a complex structure, mainly resulting from the imprint of Paleogene compressional events on a fabric inherited from the Variscan orogeny, and its Late-Variscan and Mesozoic extensional history. The prevailing foliation on the fault gouge and microbreccia is oblique (striking NW-SE) to the main E-W trace of the fault. S-C structures affecting the foliation show both sinistral and dextral shear components on NE-SW and NW-SE planes, respectively, consistent with horizontal striations on quartz rods and planar bodies. Pure reverse and normal dip-slip shear components are also inferred from S-C structures at the meso - and micro-scale, indicating a NW to N (reverse), and S-SSE (normal) transport directions. The magnetic foliation averages the orientation of the observed foliation at the outcrop scale and the magnetic lineation shows two maxima. The first maximum has an overall horizontal plunge, parallel to the strike-slip transport direction inferred from S-C structures. The second maximum is parallel to the dip direction of foliation planes and also parallelizes the reverse transport direction and occasionally the intersection lineation between S and C planes, thus revealing the strong heterogeneity and partitioning of deformation. Cataclasis and pressure-solution are the main deformation mechanisms, in agreement with clay composition and low paleotemperatures determined for the Paleogene. The presented results are consistent with Paleogene transpression occurred through strain partitioning between the main thrust separating the Catalan Coastal Range from the Ebro basin (reverse slip) and the Vallès-Penedès fault zone (sinistral strike-slip), with an important influence of dextral shear associated to the secondary, WNW-ESE, segment of the fault (either an oblique ramp or resulting from the re-activation of the Llobregat Fault). A subsequent, Neogene extensional imprint slightly modified the compressional fabrics in the fault zone. [ABSTRACT FROM AUTHOR]

To investigate the influence of foliation and structure within fault zones, synthetic fault gouges of quartz containing various thin layers of biotite were sheared in a triaxial testing system at hydrothermal conditions to acquire the faulting behaviour, along with quartz-biotite mixtures tested for comparison. First, in Series-1 experiments, gouges were sheared at a temperature of 100 °C, with effective confining pressure of 200 MPa, pore pressure of 30 MPa and loading rate of 1.22 μm/s. The results revealed that strength of quartz-biotite mixture weakened linearly with increasing biotite proportion, whereas the gouges with a biotite interlayer weakened much more significantly even for biotite content as little as ~5 wt%. Series-2 experiments were performed on simulated gouge with 15 wt% biotite interlayer at temperatures of 25–600 °C, effective normal stress of 200–230 MPa and pore pressure of 30 MPa, with stepped shear rates of 0.0488 to 1.22 μm/s. At effective normal stress of 200 MPa, the simulated foliate gouge showed a transition from velocity strengthening to velocity weakening around 200 °C, which persisted up to 600 °C. Stick-slips above 400 °C were associated with minor a – b averagely of ~−0.0006 and submicron d c values. At effective normal stress of 230 MPa and temperature of 600 °C, stick-slip motions evolved with slip distance to viscous-like behaviour at a displacement of ~3 mm. Microstructural observation on deformed samples revealed that shear deformation mainly occurred within the biotite interlayer in most cases, with surrounding quartz gouge only slightly deformed but increasingly compacted as temperature increased. Viscous-like behaviour corresponded to migration of dominant shear deformation to the quartz gouge, as indicated by the development of Riedel shears therein. Our results may help constrain fault strength and depth range of seismogenesis within fault zones associated with biotite in form of foliations and/or interconnected networks. [ABSTRACT FROM AUTHOR]

The Warramboo magnetite iron ore deposit in South Australia is predominantly hosted by gneiss. Design of the footwall slope requires an assessment of the role of foliation-parallel discontinuities as planes of weakness parallel to the slope. A review of published studies of joint spacing and joint persistence in gneiss shows spacing and persistence of foliation-parallel discontinuities relative to non-foliation-parallel discontinuities varies greatly. Foliation-parallel discontinuities form a spectrum between simply being one joint set in a blocky rock mass through to being a significant anisotropic weakness. Based on detailed analysis of the characteristics and frequency of foliation in drill core in the Warramboo magnetite deposit, it is inferred, based on analysis of spacing, that foliation-parallel discontinuities are no more frequent or persistent than other joints. Therefore the rock mass is blocky rather than anisotropic and footwall slopes steeper than the foliation dip are viable. [ABSTRACT FROM AUTHOR]

An orientable Riemannian three-manifold ( M , g ) admits an almost α -coKähler structure with g as a compatible metric if and only if M admits a foliation, defined by a unit closed 1-form, of constant mean curvature. Then, we show that a simply connected homogeneous almost α -coKähler three-manifold is either a Riemannian product of type R × S 2 ( k 2 ) , equipped with its standard coKähler structure, or it is a semidirect product Lie group G = R 2 ⋊ A R equipped with a left invariant almost α -coKähler structure. Moreover, we distinguish the several spaces of this classification by using the Gaussian curvature K G of the canonical foliation. In particular, R × S 2 ( k 2 ) is the only simply connected homogeneous almost α -coKähler three-manifolds whose canonical foliation has Gaussian curvature K G > 0 . [ABSTRACT FROM AUTHOR]

A geometric description of the first Poisson cohomology groups is given in the semilocal context, around (possibly singular) symplectic leaves. This result is based on the splitting theorems for infinitesimal automorphisms of coupling Poisson structures which describe the interaction between the tangential and transversal data of the characteristic distributions. As a consequence, we derive some criteria of vanishing of the first Poisson cohomology groups and apply the general splitting formulas to some particular classes of Poisson structures associated with singular symplectic foliations. [ABSTRACT FROM AUTHOR]

Inspired by the small sphere-limit for quasi-local energy we study local foliations of surfaces with prescribed mean curvature. Following the strategy used by Ye in [23] to study local constant mean curvature foliations we use a Lyapunov Schmidt reduction in an n + 1 dimensional manifold equipped with a symmetric 2-tensor to construct the foliations around a point, prove their uniqueness and show their nonexistence conditions. To be specific, we study two foliation conditions. First we consider constant space-time mean curvature surfaces. They are used to characterizing the center of mass in general relativity [4]. Second, we study local foliations of constant expansion surfaces [18]. [ABSTRACT FROM AUTHOR]

In this work we describe the deformation mechanisms and processes that occurred during the evolution of cataclastic deformation bands developed in the feldspar-rich conglomerates of the Rio do Peixe Basin, NE Brazil. We studied bands with different deformation intensities, ranging from single cm-thick tabular bands to more evolved clustering zones. The chemical identification of cataclastic material within deformation bands was performed using compositional mapping in SEM images, EDX and XRD analyses. Deformation processes were identified by microstructural analysis and by the quantification of comminution intensity, performed using digital image processing. The deformation bands are internally non homogeneous and developed during five evolutionary stages: (1) moderate grain size reduction, grain rotation and grain border comminution; (2) intense grain size reduction with preferential feldspar fragmentation; (3) formation of subparallel C-type slip zones; (4) formation of S-type structures, generating S-C-like fabric; and (5) formation of C′-type slip zones, generating well-developed foliation that resembles S-C-C′-type structures in a ductile environment. Such deformation fabric is mostly imparted by the preferential alignment of intensely comminuted feldspar fragments along thin slip zones developed within deformation bands. These processes were purely mechanical (i.e., grain crushing and reorientation). No clays or fluids were involved in such processes. [ABSTRACT FROM AUTHOR]

Let F ( d ; n ) be the parameter space of the family of holomorphic foliations of codimension one and degree d in P n . Gomez-Mont and Lins-Neto have shown that the Zariski closure of the set of foliations defined by a differential 1-form of type a F d G − b G d F , where F , G denote co-prime homogeneous polynomials of degrees a , b is an irreducible component of F ( a + b − 2 ; n ) . Our main result gives a formula for the degree of this component for a = 2 , b odd. [ABSTRACT FROM AUTHOR]

The Hyde-Macraes Shear Zone in southern New Zealand contains the circa 10 million ounce Macraes gold deposit, one of the larger Phanerozoic orogenic gold deposits discovered to date globally. Approximately 50% of this 10 million ounce resource is hosted by 5 major ore shoots up to 400 m wide and 1500 m long in the Frasers area at the southern end of the mine. Higher grade (>1.5 g/t Au) ore shoots are located along and immediately below the Hangingwall Shear, the principal strand of the Hyde-Macraes Shear Zone at the Frasers deposit. They typically trend parallel to the intersection of the shear and foliation in the underlying schist, commonly where the foliation dips more steeply that the overlying Hangingwall Shear. Especially thick zones of higher grade mineralised rock are located between the Hangingwall Shear and underlying second order splay shears whose position correlates with minor right-hand bends in the strike of the overlying Hangingwall Shear. Lower grade (<1.2 g/t Au), but economically significant, ore shoots are located within mineralised schists below the Hangingwall Shear. Outer margins of these lower grade ore shoots are generally parallel to the strike of the foliation in the host schist. They are most extensive where open disharmonic folding has resulted in the strike of the foliation diverging from that of the overlying Hangingwall Shear. No correlation exists between the position of any ore shoots and gently dipping jogs in the Hangingwall Shear, despite mineralisation occurring during reverse movement on the Hyde-Macraes Shear Zone. Instead the angular relationship between various strands of the Hyde-Macraes Shear Zone at Frasers and foliation in underlying schists is the most consistent structural feature likely to predict the location, extent, and orientation of ore shoots within the Frasers segment of the Hyde-Macraes Shear Zone. [ABSTRACT FROM AUTHOR]

We discuss black hole spacetimes with a geometrically defined quasi-local horizon on which the curvature tensor is algebraically special relative to the alignment classification. Based on many examples and analytical results, we conjecture that a spacetime horizon is always more algebraically special (in all of the orders of specialization) than other regions of spacetime. Using recent results in invariant theory, such geometric black hole horizons can be identified by the alignment type II or D discriminant conditions in terms of scalar curvature invariants, which are not dependent on spacetime foliations. The above conjecture is, in fact, a suite of conjectures (isolated vs dynamical horizon; four vs higher dimensions; zeroth order invariants vs higher order differential invariants). However, we are particularly interested in applications in four dimensions and especially the location of a black hole in numerical computations. [ABSTRACT FROM AUTHOR]

In this paper we study the geometric properties of a couple of mutually orthogonal foliations with complementary dimensions. We recall that from Novikov's theorem, there is no foliation of S 3 by closed curves with integrable normal bundle. Nevertheless, for S 2 k + 1 , k ≥ 2 , Novikov's theorem is not applicable. In this paper we show that on odd-dimensional unit spheres there is no umbilical foliation with integrable normal bundle and divergence free mean curvature vector. [ABSTRACT FROM AUTHOR]

The gravitational Lagrangian can be written as a summation of a bulk and a total derivative term. For some theories of gravity such as Einstein gravity, or more general Lovelock gravities, there are Lagrangian holographic relations between the bulk and the total derivative term such that the latter is fully determined by the former. However at the D → 2 & 4 limit, the bulks of Einstein or Gauss-Bonnet theories become themselves total derivatives. Performing the Kaluza-Klein reduction on Einstein and Gauss-Bonnet gravities gives rise to some two-dimensional or four-dimensional scalar-tensor theories respectively. We obtain the holographic relations for the D = 2 and D = 4 cases, which have the same form as the holographic relations in pure gravity in the foliation independent formalism. [ABSTRACT FROM AUTHOR]

Automatic hexahedral mesh generation plays a fundamental role in CAD/CAE fields, especially for IGA (isogeometric analysis). Recently, an automatic hex-mesh generation method has been proposed, which is based on the equivalence among three key concepts: colorable quadrilateral meshes, finite measured foliations and Strebel differentials. This work focuses on the computational aspect of Strebel differentials on high genus surfaces, which is based on graph-valued harmonic mapping. The algorithmic pipeline is as follows: first, the user inputs an admissible curve system, which induces a cylindric-decomposition graph; then the user specifies the lengths of the edges of the graph; third, the algorithm finds the unique harmonic map from the surface to the metric graph by a non-linear heat flow; finally, the harmonic map induces a Strebel differential, which can be further utilized to generate the hex-mesh. The method has solid theoretic foundation, which guarantees the existence and the uniqueness of the graph-valued harmonic map. The algorithm is capable of handling surfaces with complicated topologies, and producing all possible Strebel differentials on the surface. The user has full control of the combinatorial type and the geometry of the Strebel differential. The computational pipeline is automatic. The experimental results show the efficiency and efficacy of the algorithm, and demonstrate the great potential for automating the structured hexahedral mesh generation. [ABSTRACT FROM AUTHOR]

To study the space C of Kähler metrics, we can consider the following two geometric objects. One is the certain complexification Y of the symplectomorphism group, and the other one is a foliation on product Kähler manifold related to the geodesic on C . In this note we study the geometry of Y and show how the properties of the associated foliation (in particular, the holomorphicity) restrict the topology of base Kähler manifold. [ABSTRACT FROM AUTHOR]

We detail the construction of a weak Poisson bracket over a submanifold Σ of a smooth manifold M with respect to a local foliation of this submanifold. Such a bracket satisfies a weak type Jacobi identity but may be viewed as a usual Poisson bracket on the space of leaves of the foliation. We then lift this weak Poisson bracket to a weak odd Poisson bracket on the odd tangent bundle Π T M , interpreted as a weak Koszul bracket on differential forms on M . This lift is achieved by encoding the weak Poisson structure into a homotopy Poisson structure on an extended manifold, and lifting the Hamiltonian function that generates this structure. Such a construction has direct physical interpretation. For a generic gauge system, the submanifold Σ may be viewed as a stationary surface or a constraint surface, with the foliation given by the foliation of the gauge orbits. Through this interpretation, the lift of the weak Poisson structure is simply a lift of the action generating the corresponding BRST operator of the system. [ABSTRACT FROM AUTHOR]

In this paper, we use the methods of subriemannian geometry to study the dual foliation of the singular Riemannian foliation induced by isometric Lie group actions on a complete Riemannian manifold M . We show that under some conditions, the dual foliation has only one leaf. [ABSTRACT FROM AUTHOR]

The Dolbeault cohomology plays an important role in the study of some ∂ ¯ -invariants of complex and holomorphic vector bundles as ∂ ¯ -Chern classes and Atiyah classes. In this paper we generalize similar invariants and their properties in the tangential Dolbeault cohomology. More exactly, we introduce and we study tangential Atiyah classes for F -holomorphic vector bundles and ∂ ¯ -tangential Chern classes for tangentially smooth vector bundles over a manifold M endowed with a complex foliation F . Also, ∂ ¯ -tangential secondary invariants are studied following similar constructions for Lie algebroids. The notions are introduced by a global formalism that is used in the tangential theory of foliated spaces. [ABSTRACT FROM AUTHOR]

We construct explicit examples of one dimensional foliations over P n , with no proper invariant subvarieties of positive dimension, that are defined over a pure transcendental extension of Q whose transcendence degree is linear in n and independent of the degree of the foliation. [ABSTRACT FROM AUTHOR]

We study time-dependent deformations of a certain class of backgrounds in type IIB supergravity. These backgrounds are solutions of a five-dimensional consistent truncation, relevant for gauge/gravity duality, which have the form of d S 4 foliations over a fifth (radial) direction. We investigate time-dependent deformations of those solutions in the search for gravitational duals of models of glueball inflation. A particular starting ansatz enables us to find a class of analytical solutions, corresponding to an ultra-slow roll inflationary regime. This regime may play a role in understanding the low l anomaly in the power spectrum of the CMB. [ABSTRACT FROM AUTHOR]

We define Seiberg–Witten equations on closed manifolds endowed with a Riemannian foliation of codimension 4. When the foliation is taut, we show compactness of the moduli space under some hypothesis satisfied for instance by closed K -contact manifolds. Furthermore, we prove some vanishing and non-vanishing results and we highlight that the invariants may be used to distinguish different foliations on diffeomorphic manifolds. [ABSTRACT FROM AUTHOR]

The twist construction is a geometric model of T-duality that includes constructions of nilmanifolds from tori. This paper shows how one-dimensional foliations on manifolds may be used in a shear construction, which in algebraic form builds certain solvable Lie groups from Abelian ones. We discuss other examples of geometric structures that may be obtained from the shear construction. [ABSTRACT FROM AUTHOR]

With a view toward sub-Riemannian geometry, we introduce and study H-type foliations. These structures are natural generalizations of K-contact geometries which encompass as special cases K-contact manifolds, twistor spaces, 3K-contact manifolds and H-type groups. Under an horizontal Ricci curvature lower bound on these structures, we prove a sub-Riemannian diameter upper bounds and first eigenvalue estimates for the sub-Laplacian. Then, using a result by Moroianu-Semmelmann [38] , we classify the H-type foliations that carry a parallel horizontal Clifford structure. Finally, we prove an horizontal Einstein property and compute the horizontal Ricci curvature of these spaces in codimension more than 2. [ABSTRACT FROM AUTHOR]

We describe the global foliation of asymptotic lines on compact surfaces of genus greater than one, defined in R 4 as the links of the real part of the Milnor fibration of a family of polynomials. Through this analysis, we present a description of the global foliation of lines of curvature of the embeddings, defined as the images of the stereographic projection of these links into R 3. [ABSTRACT FROM AUTHOR]

We construct a type of transverse deformation of a Vaisman manifold, which preserves the canonical foliation. For this construction we only need a basic 1-form with certain properties. We show that such basic 1-forms exist in abundance. [ABSTRACT FROM AUTHOR]

In this article we provide a version of Chow's theorem for real analytic Levi-flat hypersurfaces in the complex projective space P n , n ≥ 2. More specifically, we prove that a real analytic Levi-flat hypersurface M ⊂ P n , with singular set of real dimension at most 2 n − 4 and whose Levi leaves are contained in algebraic hypersurfaces, is tangent to the levels of a rational function in P n. As a consequence, M is a semialgebraic set. We also prove that a Levi foliation on P n — a singular real analytic foliation whose leaves are immersed complex manifolds of codimension one — satisfying similar conditions — singular set of real dimension at most 2 n − 4 and all leaves algebraic — is defined by the level sets of a rational function. [ABSTRACT FROM AUTHOR]

Pointwise slant submanifolds were introduced by Chen and Garay (2012) [16] as a natural generalization of slant submanifolds. On the other hand, pointwise slant submersions were defined by Lee and Şahin (2014) [29] as a natural generalization of slant submersions. In this paper, as a generalization of pointwise slant submanifolds and pointwise slant submersions, we introduce pointwise slant Riemannian maps from almost Hermitian manifolds to Riemannian manifolds, present examples and characterizations. We investigate the geometry of foliations which are arisen from the definition of a pointwise slant Riemannian map and obtain decomposition theorems by using the existence of pointwise slant Riemannian maps. We also investigate the harmonicity of such maps and find necessary and sufficient conditions for pointwise slant Riemannian maps to be totally geodesic. Finally, we study some curvature relations in complex space forms, involving Casorati curvatures for pointwise slant Riemannian maps. [ABSTRACT FROM AUTHOR]

In this paper we generalize a theorem of M. Hilsum and G. Skandalis stating that the C ∗ -algebra of any foliation of nonzero dimension is stable. Precisely, we show that the C ∗ -algebra of a Lie groupoid is stable whenever the groupoid has no orbit of dimension zero. We also prove an analogous theorem for singular foliations for which the holonomy groupoid as defined by I. Androulidakis and G. Skandalis is not Lie in general. [ABSTRACT FROM AUTHOR]

The aim of this paper is to classify compact, simply connected Kähler manifolds which admit totally geodesic, holomorphic complex homothetic foliations by curves. [ABSTRACT FROM AUTHOR]

We show that every circle on a complex hyperbolic space except unbounded circles of complex torsion ±1 which have two distinct points at infinity can be regarded as a trajectory for some Sasakian magnetic field on some totally η -umbilic real hypersurface. Our study explains why the lamination on the moduli space of circles on a complex hyperbolic space has singularities only on the leaf of circles of complex torsion ±1. [ABSTRACT FROM AUTHOR]

We study Bakry-Émery type estimates for the Laplace–Beltrami operator of a totally geodesic foliation. In particular, we are interested in situations for which the Γ 2 operator may not be bounded from below but the horizontal Bakry-Émery curvature is. As we prove it, under a bracket generating condition, this weaker condition is enough to imply several functional inequalities for the heat semigroup including the Wang–Harnack inequality and the log-Sobolev inequality. We also prove that, under proper additional assumptions, the generalized curvature dimension inequality introduced by Baudoin and Garofalo (2015) is uniformly satisfied for a family of Riemannian metrics that converge to the sub-Riemannian one. [ABSTRACT FROM AUTHOR]

In this note we give a characterization of taut Riemannian foliations using the transverse divergence. This result turns out to be a convenient tool in the case of some standard examples. Furthermore, we show that a classical tautness result of Haefliger can be obtained in our particular setting as a straightforward consequence. In the final part of the paper we obtain a tautness characterization for transversally oriented foliations with dense leaves. [ABSTRACT FROM AUTHOR]

The earliest pervasive deformation events in the Archaean Eastern Goldfields Superterrane of the Yilgarn Craton are partly obscured by subsequent deformation and pre-date rocks exposed over large areas. As a result, uncertainty exists about the nature, timing and role that early deformation events played in controlling regional architecture and gold mineralization in this world-class metallogenic province. The Daisy Milano mining camp in the southern Kurnalpi Terrane has opened up new exposures in relatively old geological sequences, which record structures that pre-date the main ENE-WSW D 2 shortening event. These D 1 structures represent the earliest pervasive deformation events in the Eastern Goldfields Superterrane and are characterized by a pervasive bedding-parallel foliation (S 1 //S 0 ) that accommodated NW-verging thrusting. The same foliation planes also accommodated normal movement verging to the SE, interpreted to indicate periods of stress relaxation. Alternatively, thrusting and normal movement could have been contemporaneous, defining a pure shear event. D 1 structures are overprinted by steeply dipping, NNW-striking foliation planes that are axial planar to folds and local crenulation cleavage (F 2 folds). This second foliation is parallel to and overprints post-D 1 , sub-vertical, plagioclase-phyric tonalite dikes, at the margins of which the Daisy Milano gold deposits formed. Deposits occur stratigraphically below moderately SW-dipping ultramafic layers that are sub-parallel to the early thrust foliation and acted as impermeable caps to mineralizing fluids. Mineralization occurs also in NE-SW to ENE-WSW quartz veins in the hinge zones of open and gently plunging F 2 folds in a competent dolerite sill. Mineralization was either synchronous with or post-dated D 2 . Two samples of tonalite dikes at Daisy Milano were analyzed using zircon LA-ICP-MS U–Pb geochronology and yielded ages of 2687 ± 7 Ma and 2676 ± 9 Ma, placing an upper bound on the timing of D 2 and gold mineralization at Daisy Milano, and a lower bound on the timing of D 1 . [ABSTRACT FROM AUTHOR]

We study Riemannian foliations whose transverse Levi-Civita connection ∇ has special holonomy. In particular, we focus on the case where Hol ( ∇ ) is contained either in SU ( n ) or in Sp ( n ) . We prove a Weitzenböck formula involving complex basic forms on Kähler foliations and we apply this formula for pointing out some properties of transverse Calabi–Yau structures. This allows us to prove that links provide examples of compact simply-connected contact Calabi–Yau manifolds. Moreover, we show that a simply-connected compact manifold with a Kähler foliation admits a transverse hyper-Kähler structure if and only if it admits a compatible transverse hyper-Hermitian structure. This latter result is the ‘‘foliated version” of a theorem proved by Verbitsky in [46] . In the last part of the paper we adapt our results to the Sasakian case, showing in addition that a compact Sasakian manifold has trivial transverse holonomy if and only if it is a compact quotient of the Heisenberg Lie group. [ABSTRACT FROM AUTHOR]