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In order to generalise semi-invariant Riemannian maps, Sahin first introduced the idea of "Generic Riemannian maps". We extend the idea of generic Riemannian maps to the case in which the total manifold is a nearly Kaehler manifold. We study the integrability conditions for the horizontal distribution although vertical distribution is always integrable. We also study the geometry of foliations of two distributions and obtain the necessary and sufficient condition for generic Riemannian maps to be totally geodesic. Additionally, we study the generic Riemannian map with umbilical fibers. [ABSTRACT FROM AUTHOR]

We investigate the foliation defined by the kernel of an exact presymplectic form dα of rank 2n on a (2n + r)-dimensional closed manifold M. For r = 2, we prove that the foliation has at least two leaves which are homeomorphic to a 2-dimensional torus, if M admits a locally free T²-action which preserves dα and satisfies that the function α(Z2) is constant, where Z1,Z2 are the infinitesimal generators of the T²-action. We also give its generalization for r ≥ 1. [ABSTRACT FROM AUTHOR]

Given a uniform foliation by Gromov hyperbolic leaves on a 3-manifold, we show that the action of the fundamental group on the universal circle is minimal and transitive on pairs of different points. We also prove two other results: we prove that general uniform Reebless foliations are R-covered and we give a new description of the universal circle of R-covered foliations with Gromov hyperbolic leaves in terms of the JSJ decomposition of M. [ABSTRACT FROM AUTHOR]

Androulidakis-Skandalis (2009) showed that every singular foliation has an associated topological groupoid, called holonomy groupoid. In this note, we exhibit some functorial properties of this assignment: if a foliated manifold (M, FM) is the quotient of a foliated manifold (P, FP) along a surjective submersion with connected fibers, then the same is true for the corresponding holonomy groupoids. For quotients by a Lie group action, an analogue statement holds under suitable assumptions, yielding a Lie 2-group action on the holonomy groupoid. [ABSTRACT FROM AUTHOR]

The aim of this series of papers is to generalize the ambient approach of Duval et al. regarding the embedding of Galilean and Carrollian geometries inside gravitational waves with parallel rays. In this paper (Paper I), we propose a generalization of the embedding of torsionfree Galilean and Carrollian manifolds inside larger classes of gravitational waves. On the Galilean side, the quotient procedure of Duval et al. is extended to gravitational waves endowed with a lightlike hypersurface-orthogonal Killing vector field. This extension is shown to provide the natural geometric framework underlying the generalization by Lichnerowicz of the Eisenhart lift. On the Carrollian side, a new class of gravitational waves – dubbed Dodgson waves – is introduced and geometrically characterized. Dodgson waves are shown to admit a lightlike foliation by Carrollian manifolds and furthermore to be the largest subclass of gravitational waves satisfying this property. This extended class allows us to generalize the embedding procedure to a larger class of Carrollian manifolds that we explicitly identify. As an application of the general formalism, (Anti) de Sitter spacetime is shown to admit a lightlike foliation by codimension one (A)dS Carroll manifolds. [ABSTRACT FROM AUTHOR]

We study the deformation theory of pre-symplectic structures, that is, closed 2-forms of fixed rank. The main result is a parametrization of nearby deformations of a given pre-symplectic structure in terms of an |$L_{\infty }$| -algebra, which we call the Koszul |$L_{\infty }$| -algebra. This |$L_{\infty }$| -algebra is a cousin of the Koszul dg Lie algebra associated to a Poisson manifold. In addition, we show that a quotient of the Koszul |$L_{\infty }$| -algebra is isomorphic to the |$L_{\infty }$| -algebra that controls the deformations of the underlying characteristic foliation. Finally, we show that the infinitesimal deformations of pre-symplectic structures and of foliations are both obstructed. [ABSTRACT FROM AUTHOR]

We study semi-invariant Riemannian submersions from a nearly Kaehler manifold to a Riemannian manifold. It is well known that the vertical distribution of a Riemannian submersion is always integrable therefore, we derive condition for the integrability of horizontal distribution of a semi-invariant Riemannian submersion and also investigate the geometry of the foliations. We discuss the existence and nonexistence of semi-invariant submersions such that the total manifold is a usual product manifold or a twisted product manifold. We establish necessary and sufficient conditions for a semi-invariant submersion to be a totally geodesic map. Finally, we study semi-invariant submersions with totally umbilical fibers. [ABSTRACT FROM AUTHOR]

If Y is a closed orientable graph manifold, we show that Y admits a coorientable taut foliation if and only if Y is not an L-space. Combined with previous work of Boyer and Clay, this implies that Y is an L-space if and only if π1(Y) is not left-orderable. [ABSTRACT FROM AUTHOR]

We introduce the notion of an asymptotically Poincaré family of surfaces in an end of a quasi-Fuchsian manifold. We show that any such family gives a foliation of an end by asymptotically parallel convex surfaces, and that the asymptotic behavior of the first and second fundamental forms determines the projective structure at infinity. As an application, we establish a conjecture of Labourie from [J. London Math. Soc. 45 (1992), pp. 549-565] regarding constant Gaussian curvature surfaces. We also derive consequences for constant mean curvature surfaces. [ABSTRACT FROM AUTHOR]

In 3-dimensional manifolds, we prove that generically in Diff1m(M3), the existence of a minimal expanding invariant foliation implies stable Bernoulliness. [ABSTRACT FROM AUTHOR]

For an oriented manifold M whose dimension is less than 4, we use the contractibility of certain complexes associated to its submanifolds to cut M into simpler pieces in order to do local to global arguments. In particular, in these dimensions, we give a different proof of a deep theorem of Thurston in foliation theory that says the natural map between classifying spaces BHomeoδ(M) → BHomeo(M) induces a homology isomorphism where Homeoδ(M) denotes the group of homeomorphisms of M made discrete. Our proof shows that in low dimensions, Thurston's theorem can be proved without using foliation theory. Finally, we show that this technique gives a new perspective on the homotopy type of homeomorphism groups in low dimensions. In particular, we give a different proof of Hacher's theorem that the homeomorphism groups of Haken 3-manifolds with boundary are homotopically discrete without using his disjunction techniques. [ABSTRACT FROM AUTHOR]

Let X be a compact connected Kähler manifold of dimension ≥ 3 and L a C ∞ Levi flat hypersurface in X. Then the normal bundle to the Levi foliation does not admit a Hermitian metric with positive curvature along the leaves. This represents an answer to a conjecture of Marco Brunella. [ABSTRACT FROM AUTHOR]

A result due to M.W. Hirsch states that most competitive maps admit a carrying simplex, i.e., an invariant hypersurface of codimension one which attracts all nontrivial orbits. The common approach in the study of these maps is to focus on the dynamical behavior on the carrying simplex. However, this manifold is normally non-smooth. Therefore, not every tool coming from Differential Geometry can be applied. In this paper we prove that the restriction of the map to the carrying simplex in a neighborhood of an interior fixed point is topologically conjugate to the restriction of the map to its pseudo-unstable manifold by an invariant foliation. This implies that the linearization techniques are applicable for studying the local dynamics of the interior fixed points on the carrying simplex. We further construct the stable and unstable manifolds on the carrying simplex. Our results give partial responses to Hirsch's problem regarding the smoothness of the carrying simplex. We discuss some applications in classical models of population dynamics. [ABSTRACT FROM AUTHOR]

This is the first in a series of papers dedicated to the study of Poisson manifolds of compact types (PMCTs). This notion encompasses several classes of Poisson manifolds defined via properties of their symplectic integrations. In this first paper we establish some fundamental properties and constructions of PMCTs. For instance, we show that their Poisson cohomology behaves very much like the de Rham cohomology of a compact manifold (Hodge decomposition, non-degenerate Poincaré duality pairing, etc.) and that the Moser trick can be adapted to PMCTs. More important, we find unexpected connections between PMCTs and symplectic topology: PMCTs are related with the theory of Lagrangian fibrations and we exhibit a construction of a non-trivial PMCT related to a classical question on the topology of the orbits of a free symplectic circle action. In subsequent papers, we will establish deep connections between PMCTs and integral affine geometry, Hamiltonian G-spaces, foliation theory, orbifolds, Lie theory and symplectic gerbes. [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]

We combine classic stability results for foliations with recent results on deformations of Lie groupoids and Lie algebroids to provide a cohomological characterization for rigidity of compact Hausdorff foliations on compact manifolds. [ABSTRACT FROM AUTHOR]

We construct smooth fiber bundles such that the fibers are exotic tori and the total space has finite abelian fundamental group. This gives examples of a Riemannian foliation on a closed manifold whose leaves are exotic tori and whose total space has finite abelian fundamental group. [ABSTRACT FROM AUTHOR]

Let M be a 2n-dimensional smooth manifold associated with the structure of symplectic pair which is a pair of closed 2-forms of constant ranks with complementary kernel foliations. Let Q ⊂ M be a codimension 2 compact submanifold. We show some sufficient and necessary conditions on the existence of the structure of contact pair (α, β) on Q, which is a pair of 1-forms of constant classes whose characteristic foliations are transverse and complementary such that α and β restrict to contact forms on the leaves of the characteristic foliations of β and α, respectively. This is a generalization of the neighborhood theorem for contact-type hypersurfaces in symplectic topology. [ABSTRACT FROM AUTHOR]

Classification results for complex Riemannian foliations are obtained. For open subsets of irreducible Hermitian symmetric spaces of compact type, where one has explicit control over the curvature tensor, we completely classify such foliations by studying the infinitesimal model associated with the canonical connection. We also establish results for symmetric spaces of noncompact type and a general rigidity result for any irreducible Kähler manifold. [ABSTRACT FROM AUTHOR]

In general relativity, there have been a number of successful constructions for asymptotically flat metrics with a certain background foliation. In particular, Lin (Calc Var 49(3-4):1309-1335, 2014) used a foliation by the Ricci flow on 2-spheres to establish an asymptotically flat extension, and Sormani and Lin (Ann Henri Poincaré 17(10):2783-2800, 2016) proved useful results with this extension. In this paper, we construct asymptotically hyperbolic 3-metrics with the Ricci flow foliation inspired by Lin's work. We also study the rigid case when the Hawking mass of the inner surface of the manifold agrees with its total mass as in Sormani and Lin (2016). [ABSTRACT FROM AUTHOR]

We introduce a notion of equivalence for singular foliations—understood as suitable families of vector fields—that preserves their transverse geometry. Associated with every singular foliation, there is a holonomy groupoid, by the work of Androulidakis-Skandalis. We show that our notion of equivalence is compatible with this assignment, and as a consequence, we obtain several invariants. Further, we show that it unifies some of the notions of transverse equivalence for regular foliations that appeared in the 1980s. [ABSTRACT FROM AUTHOR]

In this paper we make three remarks about metallic Riemannian manifolds. Firstly we give an integrability condition using a Codazzi type equation, then we consider another metallic structure on a metallic Riemannian manifold and we study the mixed twin metric associated to them. Finally, we show that a Riemannian foliated manifold has a natural (p, q)-metallic structure and every leaf of the foliation carries a metallic structure which is always integrable. [ABSTRACT FROM AUTHOR]

We show that any two disjoint crooked planes in ℝ 3 are leaves of a crooked foliation. This answers a question asked by Charette and Kim [V. Charette and Y. Kim, Foliations of Minkowski 2 + 1 spacetime by crooked planes, Int. J. Math.25(9) (2014) 1450088.]. [ABSTRACT FROM AUTHOR]

The basic cohomology of a complete Riemannian foliation with all leaves closed is the cohomology of the leaf space. In this paper we introduce various methods to compute the basic cohomology in the presence of both closed and non-closed leaves in the simply-connected case (or more generally for Killing foliations): We show that the total basic Betti number of the union C of the closed leaves is smaller than or equal to the total basic Betti number of the foliated manifold, and we give sufficient conditions for equality. If there is a basic Morse–Bott function with critical set equal to C, we can compute the basic cohomology explicitly. Another case in which the basic cohomology can be determined is if the space of leaf closures is a simple, convex polytope. Our results are based on Molino's observation that the existence of non-closed leaves yields a distinguished transverse action on the foliated manifold with fixed point set C. We introduce equivariant basic cohomology of transverse actions in analogy to equivariant cohomology of Lie group actions enabling us to transfer many results from the theory of Lie group actions to Riemannian foliations. The prominent role of the fixed point set in the theory of torus actions explains the relevance of the set C in the basic setting. [ABSTRACT FROM AUTHOR]

In this paper, we study the non-trivial warped product pseudo-slant submanifold of a nearly cosymplectic manifold such that the extrinsic sphere is slant immersion. outset of the considering, we also show some integrability and totally geodesic foliations results for the distributions of pseudo-slant submanifold. Next, we demonstrate the characterization theorem for the existence of warped product isometrically immersed into nearly cosymplectic manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we develop symplectic Hodge theory on transversely symplectic foliations. In particular, we establish the symplectic dδ-lemma for any such foliations with the (transverse) s-Lefschetz property. As transversely symplectic foliations include many geometric structures, such as contact manifolds, co-symplectic manifolds, symplectic orbifolds and symplectic quasifolds as special examples, our work provides a unifying treatment of symplectic Hodge theory in these geometries. As an application, we show that on compact K-contact manifolds, the s-Lefschetz property implies a general result on the vanishing of cup products, and that the cup length of a 2n + 1 dimensional compact K-contact manifold with the (transverse) s-Lefschetz property is at most 2n - s. For any even integer s≥2, we also apply our main result to produce examples of K-contact manifolds that are s-Lefschetz, but not (s + 1)-Lefschetz. [ABSTRACT FROM AUTHOR]

We construct special classes of totally geodesic almost regular foliations, namely, complex radial foliations in Hermitian manifolds and quaternionic radial foliations in quaternionic Kähler manifolds, and we give criteria for their harmonicity and minimality. Then examples of these foliations on complex and quaternionic space forms, which are harmonic and minimal, are presented. [ABSTRACT FROM AUTHOR]

In this paper we investigate the spectral sequence associated with a Riemannian foliation which arises naturally on a Vaisman manifold. Using the Betti numbers of the underlying manifold we establish a lower bound for the dimension of some terms of this cohomological object. This way we obtain cohomological obstructions for two-dimensional foliations to be induced from a Vaisman structure. We show that if the foliation is quasi-regular the lower bound is realized. In the final part of the paper we discuss two examples. [ABSTRACT FROM AUTHOR]

Let M be a compact manifold equipped with a pair of complementary foliations, say horizontal and vertical. In Catuogno et al. (Stoch Dyn 13(4):1350009, 2013) it is shown that, up to a stopping time τ, a stochastic flow of local diffeomorphisms φt in M can be written as a Markovian process in the subgroup of diffeomorphisms which preserve the horizontal foliation composed with a process in the subgroup of diffeomorphisms which preserve the vertical foliation. Here, we discuss topological aspects of this decomposition. The main result guarantees the global decomposition of a flow if it preserves the orientation of a transversely orientable foliation. In the last section, we present an Itô-Liouville formula for subdeterminants of linearised flows. We use this formula to obtain sufficient conditions for the existence of the decomposition for all t≥0. [ABSTRACT FROM AUTHOR]

We show how certain topological properties of co-Kähler manifolds derive from those of the Kähler manifolds which construct them. In particular, we show that the existence of parallel forms on a co-Kähler manifold reduces the computation of cohomology from the de Rham complex to certain amenable sub-cdga's defined by geometrically natural operators derived from the co-Kähler structure. This provides a simpler proof of the formality of the foliation minimal model in this context. [ABSTRACT FROM AUTHOR]

We show that closed aspherical manifolds supporting an affine structure, whose holonomy map is injective and contains a pure translation, must have vanishing simplicial volume. As a consequence, these manifolds have zero Euler characteristic, satisfying the Chern Conjecture. Along the way, we provide a simple cohomological criterion for aspherical manifolds with normal amenable subgroups of π1 to have vanishing simplicial volume. This answers a special case of a question due to Lück. [ABSTRACT FROM AUTHOR]

In this paper we investigate the relationships between closed AdS 3-manifolds and Higgs bundles. We have a new way to construct AdS structures that allows us to see many of their properties explicitly, for example we can recover the very recent formula by Tholozan for their volume. We give natural foliations of the AdS structure with time-like geodesic circles and we use these circles to construct equivariant minimal immersions of the Poincaré disc into the Grassmannian of time-like 2-planes of ℝ2,2. [ABSTRACT FROM AUTHOR]

We study the regularization of an oriented 1-foliation F on M ∖ Σ where M is a smooth manifold and Σ ⊂ M is a closed subset, which can be interpreted as the discontinuity locus of F . In the spirit of Filippov's work, we define a sliding and sewing dynamics on the discontinuity locus Σ as some sort of limit of the dynamics of a nearby smooth 1-foliation and obtain conditions to identify whether a point belongs to the sliding or sewing regions. [ABSTRACT FROM AUTHOR]

In this paper we prove that the anti-canonical bundle of a holomorphic foliation F on a complex projective manifold cannot be nef and big if either F is regular, or F has a compact leaf. Then we address codimension one regular foliations whose anti-canonical bundle is nef with maximal Kodaira dimension. [ABSTRACT FROM AUTHOR]

We establish the existence of a stable foliation in the vicinity of a traveling front solution for systems of reaction diffusion equations in one space dimension that arise in the study of chemical reactions models and solid fuel combustion. In this way we complement the orbital stability results from earlier papers by A. Ghazaryan, S. Schecter and Y. Latushkin. The essential spectrum of the differential operator obtained by linearization at the front touches the imaginary axis. In spaces with exponential weights, one can shift the spectrum to the left. We study the nonlinear equation on the intersection of the unweighted and weighted spaces. Small translations of the front form a center unstable manifold. For each small translation we prove the existence of a stable manifold containing the translated front and show that the stable manifolds foliate a small ball centered at the front. [ABSTRACT FROM AUTHOR]

In this paper, we describe all invariant distributions of nondegenerate bi-Hamiltonian structures and investigate their integrability in the neighborhood of a generic point. [ABSTRACT FROM AUTHOR]

We study the foliation defined by a closed 1-form on a connected smooth closed orientable manifold. We call such a foliation compactifiable if all its leaves are closed in the complement of the singular set. In this paper, we give sufficient conditions for compactifiability of the foliation in homological terms. We also show that under these conditions, the foliation can be defined by closed 1-forms with the ranks of their groups of periods in a certain range. In addition, we describe the structure of the group generated by the homology classes of all compact leaves of the foliation. [ABSTRACT FROM AUTHOR]

We study the propagation of bosonic strings in singular target space-times. For describing this, we assume this target space to be the quotient of a smooth manifold M by a singular foliation $${{\mathcal {F}}}$$ on it. Using the technical tool of a gauge theory, we propose a smooth functional for this scenario, such that the propagation is assured to lie in the singular target on-shell, i.e., only after taking into account the gauge-invariant content of the theory. One of the main new aspects of our approach is that we do not limit $${{\mathcal {F}}}$$ to be generated by a group action. We will show that, whenever it exists, the above gauging is effectuated by a single geometrical and universal gauge theory, whose target space is the generalized tangent bundle $$TM\oplus T^*M$$ . [ABSTRACT FROM AUTHOR]

In this paper, we show the convexity of the image of a moment map on a transverse symplectic manifold equipped with a torus action under a certain condition. We also study properties of moment maps in the case of transverse Kähler manifolds. As an application, we give a positive answer to the conjecture posed by Cupit-Foutou and Zaffran. [ABSTRACT FROM AUTHOR]

In this paper we classify codimension 1 Mukai foliations on complex projective manifolds. [ABSTRACT FROM AUTHOR]

We show that the properties of admitting a co-oriented taut foliation and having a left-orderable fundamental group are equivalent for rational homology 3-sphere graph manifolds and relate them to the property of not being a Heegaard–Floer L-space. This is accomplished in several steps. First we show how to detect families of slopes on the boundary of a Seifert fibred manifold in four different fashions—using representations, using left-orders, using foliations, and using Heegaard–Floer homology. Then we show that each method of detection determines the same family of detected slopes. Next we provide necessary and sufficient conditions for the existence of a co-oriented taut foliation on a graph manifold rational homology 3-sphere, respectively a left-order on its fundamental group, which depend solely on families of detected slopes on the boundaries of its pieces. The fact that Heegaard–Floer methods can be used to detect families of slopes on the boundary of a Seifert fibred manifold combines with certain conjectures in the literature to suggest an L-space gluing theorem for rational homology 3-sphere graph manifolds as well as other interesting problems in Heegaard–Floer theory. [ABSTRACT FROM AUTHOR]

We prove that the characteristic foliation [ABSTRACT FROM AUTHOR]

We prove the strong Weinstein conjecture for closed contact manifolds that appear as the concave boundary of a symplectic cobordism admitting an essential local foliation by holomorphic spheres. [ABSTRACT FROM AUTHOR]

In an earlier paper, we studied manifolds M endowed with a generalized F structure $$\Phi \in \mathrm{End}(TM\oplus T^*M)$$ , skew-symmetric with respect to the pairing metric, such that $$\Phi ^3+\Phi =0$$ . Furthermore, if $$\Phi $$ is integrable (in some well-defined sense), $$\Phi $$ is a generalized CRF structure. In the present paper, we study quasi-classical generalized F and CRF structures, which may be seen as a generalization of the holomorphic Poisson structures (it is well known that the latter may also be defined via generalized geometry). The structures that we study are equivalent to a pair of tensor fields $$(A\in \mathrm{End}(TM),\pi \in \wedge ^2TM)$$ , where $$A^3+A=0$$ and some relations between A and $$\pi $$ hold. We establish the integrability conditions in terms of $$(A,\pi )$$ . They include the facts that A is a classical CRF structure, $$\pi $$ is a Poisson bivector field and $$\mathrm{im}\,A$$ is a (non)holonomic Poisson submanifold of $$(M,\pi )$$ . We discuss the case where either $$\mathrm{ker}\,A$$ or $$\mathrm{im}\,A$$ is tangent to a foliation and, in particular, the case of almost contact manifolds. Finally, we show that the dual bundle of $$\mathrm{im}\,A$$ inherits a Lie algebroid structure and we briefly discuss the Poisson cohomology of $$\pi $$ , including an associated spectral sequence and a Dolbeault type grading. [ABSTRACT FROM AUTHOR]

Let Z be a non-compact two-dimensional manifold obtained from a family of open strips R × (0, 1) with boundary intervals by gluing those strips along their boundary intervals. Every such strip has a foliation into parallel lines R × t, t 2∈ (0, 1), and boundary intervals, where we get a foliation Δ on all of Z. Many types of foliations on surfaces with leaves homeomorphic to the real line have such "stripedِ" structure. That fact was discovered by W. Kaplan (1940-41) for foliations on the plane R2 by level-set of pseudo-harmonic functions R2 → R without singularities. Previously, the first two authors studied the homotopy type of the group H(Δ) of homeomorphisms of Z sending leaves of Δ into leaves, and shown that except for two cases the identity path component H0 (Δ) of H(Δ) is contractible. The aim of the present paper is to show that the quotient H(Δ)/H0(Δ) can be identied with the group of automorphisms of a certain graph with additional structure encoding the ُ combinatorics"ِ of gluing. [ABSTRACT FROM AUTHOR]

We introduce slant lightlike submersions from an indefinite almost Hermitian manifold into a lightlike manifold. We establish the existence theorems for these submersions and investigate the necessary and sufficient conditions for the leaves of the distributions to be totally geodesic foliations in indefinite almost Hermitian manifolds. [ABSTRACT FROM AUTHOR]

A semiholomorphic foliation of type (n, d) is a differentiable real manifold X of dimension 2n + d, foliated by complex leaves of complex dimension n. In the present work, we introduce an appropriate notion of pseudoconvexity (and consequently, q-completeness) for such spaces, given by the interplay of the usual pseudoconvexity along the leaves, and the positivity of the transversal bundle. For 1-complete real analytic semiholomorphic foliations, we obtain a vanishing theorem for the CR cohomology, which we use to show an extension result for CR functions on Levi flat hypersurfaces and an embedding theorem in CN. In the compact case, we introduce a notion of weak positivity for the transversal bundle, which allows us to construct a real analytic embedding in CPN. [ABSTRACT FROM AUTHOR]

For an oriented irreducible 3-manifold M with non-empty toroidal boundary, we describe how sutured Floer homology (SFH) can be used to determine all fibered classes in H¹(M). Furthermore, we show that the SFH of a balanced sutured manifold (M,γ) detects which classes in H1(M) admit a taut depth one foliation such that the only compact leaves are the components of R(γ). The latter had been proved earlier by the first author under the extra assumption that H2(M)=0. The main technical result is that we can obtain an extremal Spinc-structure s (i.e., one that is in a `corner' of the support of SFH) via a nice and taut sutured manifold decomposition even when H2(M)≠0, assuming the corresponding group SFH(M,γ,s) has non-trivial Euler characteristic. [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]