The paper explores affine Hirsch foliations on 3-manifolds and classifies their existence.
problem Classifying the existence of affine Hirsch foliations on 3-manifolds.
method Analyzing and constructing 3-manifolds with specific foliations using exchangeable braided links.
result Every closed orientable 3-manifold admits 0, 1, or 2 affine Hirsch foliations, and every case is possible.
Proves Teichmüller space of Hirsch foliation homeomorphic to closed curves.
problem Minimal foliation of a closed 3-manifold by hyperbolic surfaces.
method Proves homeomorphism and uses properties of hyperbolic metrics and homeomorphisms.
result Structure group of bundle is contractible.
Study of affine and projective structures on foliated complex manifolds.
problem Formalizing and analyzing affine and projective structures on foliations.
method Formalizing concepts, providing local normal forms, proving index formulae, classifying structures.
result Compact algebraic manifolds of even dimension do not admit foliated projective structures.
The paper studies affine manifolds with linear foliations and their topological properties.
problem Characterizing the topology of affine manifolds with linear foliations.
method Analyzing the structure of affine manifolds and their foliations, proving topological properties.
result Compact affine manifolds with linear foliations are homeomorphic to the torus under certain conditions.
New theorem extends Hadamard's to transversely affine geometry.
problem Transversely affine foliations and their properties.
method Introduced and investigated novel transversely affine foliations, extending Hadamard's theorem.
result Transversely affine version of Hadamard's theorem for foliations.
We study holomorphic foliations with an affine homogeneous transverse structure. We give a friendly characterization of the case of transversely affine foliations in terms of matrix valued pairs of differential forms. This leads naturally to the study of the case of foliations with singularities. A first extension theo…
Study affinely transverse foliations in sphere bundles, finding bounds and vanishing conditions.
problem Understanding affine transverse foliations in sphere bundles and their properties.
method Provided upper bounds for the Euler number and a new proof for vanishing conditions under amenable fundamental group.
result Upper bounds and vanishing conditions for the Euler number of sphere bundles.
The abstract proves a Leray-Hirsch theorem and blow-up formula for Dolbeault cohomology.
problem Cohomology on complex manifolds, especially non-compact ones.
method Proves a Leray-Hirsch theorem and an explicit blow-up formula.
result Explicit blow-up formula for Dolbeault cohomology on complex manifolds.
Study rigid Lie affine foliations on compact manifolds.
problem Cohomological criterion for rigidity of Lie foliations.
method Detailed study of cohomology groups, Morse-Novikov cohomology.
result Many examples of rigid Lie affine foliations on compact manifolds.
In 3D affine space, unique foliation of domains by surfaces with constant Gaussian curvature.
problem Defining and studying regular domains in affine space.
method Analysis of Monge-Ampère equation with boundary condition.
result Every proper regular domain in 3D affine space is uniquely foliated by surfaces with constant Gaussian curvature.
Complex foliation cohomology is shown to be simple.
problem Analyzing cohomology of a specific complex foliation.
method Explicitly computed the Dolbeault cohomology in degree 1.
result Foliated Dolbeault cohomology in degree 1 is isomorphic to C.
Explicitly determined foliated cohomology of affine Reeb flow on Hopf manifold.
problem Determine the foliated cohomology of the affine Reeb flow on the Hopf manifold.
method Explicit calculation and analysis of the cohomology space and its dual.
result The space HF1(M) contains obstructions to solving the cohomological equation. Defines foliation criterion for dense isoperiodic leaves in rank 1 affine orbifolds.
problem Dynamics of isoperiodic leaves in rank 1 affine invariant suborbifolds.
method Defines foliation FM and establishes density criterion.
result Establishes criterion for density of isoperiodic leaves.
Study on completeness of foliations and null Killing fields in Lorentzian manifolds.
problem Completeness of foliations and null Killing fields in Lorentzian manifolds.
method Analyzes different geometric settings and conditions for completeness, including totally geodesic lightlike foliations and specific affine structures.
result Characterizes completeness for null Killing fields and specific affine structures, and provides non-complete examples.
The paper shows measures equidistribute on affine submanifolds with a rate.
problem Understanding equidistribution of measures on affine invariant submanifolds.
method Analyzing unstable foliations and using results from homogeneous dynamics.
result Measures of large dimension equidistribute on affine invariant submanifolds with an effective rate.
The leaf space of a Killing Riemannian foliation is a diffeological quasifold.
problem Understanding the structure of leaf spaces of Riemannian foliations.
method Analyzing the holonomy groupoid and using diffeological spaces.
result The leaf space of a Killing Riemannian foliation is a diffeological quasifold.
Identifies holonomy of affine surfaces via meromorphic connections.
problem Holonomy of complex affine surfaces.
method Moduli space of complex affine surfaces and meromorphic connections.
result Holonomy map is a submersion for non-finite-area translation surfaces.
Study shows dynamics of rank 1 orbifolds in flat surfaces.
problem Characterize dynamics of rank 1 affine invariant orbifolds.
method Analyzes M-isoperiodic foliations and their ergodic properties.
result Leaves of the isoperiodic foliation are either all closed or all dense.
Study jets of flat partial connections in foliations.
problem Characterize and understand flat partial connections in foliations.
method Define and apply jets to flat partial connections in smooth foliations and locally free sheaves, focusing on codimension one and arbitrary codimension foliations.
result Define and apply jets to characterize transversely affine and projective structures in foliations.
Study of affine surfaces and their Veech groups, generalizing Veech groups to new surfaces.
problem Understanding dynamics on affine surfaces and their Veech groups.
method Definition and analysis of affine surfaces, generalization of Veech groups to affine surfaces.
result Structure result about Veech groups of affine surfaces.
Study Teichmüller dynamics and dilation tori properties, proving almost all vertical foliations are Morse-Smale.
problem Understanding the coarse geometry of dilation tori and Teichmüller flow dynamics.
method Analysis of moduli space of dilation tori, Teichmüller flow action, and piecewise affine circle homeomorphisms.
result Vertical foliations of dilation tori are almost always Morse-Smale.
The space of broken hyperbolic structures generalizes the Teichmüller space of a punctured surface, and the space of projectivized broken measured foliations (equivalently, the space of projectivized affine foliations) generalizes the space of projectivized measured foliations. Just as projectivized measured foliations…
Abstract: Study of 2-knot groups with restrictions on normal subgroups.
problem Characterizing 2-knot groups based on their normal subgroups.
method Analyzing PD_4-complexes and using properties of π_1(X).
result Characterization of 2-knot groups based on their normal subgroups.
Study of foliations' geometric and topological structures.
problem Analyzing the geometric and topological properties of transversely affine foliations.
method Attach holonomy group and quotient stack, identify reparametrisations, classify them, and study the Kato-Nakayama space.
result Holonomy group controls the geometric part, while the Kato-Nakayama space captures the topological and dynamical aspects.
Quantum theory for gerbes connects stack invariants.
problem Quantum theory of gerbes over stacks.
method Proves structure result on Gromov-Witten theory.
result Gromov-Witten invariants of gerbes relate to stack invariants.
In our book on cohomological methods in transformation groups the minimal Hirsch-Brown model was used to good effect. The construction there, however, was rather abstract. Here, for smooth compact connected Lie group actions on smooth closed manifolds, we give a much more explicit construction of the minmal Hirsch-Brow…
Classifies actions on complex space forms with Lagrangian orbits.
problem Classifying actions on complex space forms with Lagrangian orbits.
method Classifies holomorphic isometric actions on complex space forms.
result Only examples are Lagrangian affine subspace foliations of complex Euclidean spaces and Lagrangian horocycle foliations of complex hyperbolic spaces.
The study shows how to foliate convex hypersurfaces in affine space with constant curvature.
problem Finding convex hypersurfaces with constant Gauss-Kronecker curvature in affine space.
method Solving a Monge-Ampère equation with specific boundary conditions.
result Regular domains in affine space are foliated by complete convex hypersurfaces with constant Gauss-Kronecker curvature.
Affine connections linked to Riccati distributions on compact surfaces.
problem Understanding affine structures on complex compact surfaces.
method Established a correspondence between affine connections and Riccati distributions.
result One-to-one correspondence between affine structures and Riccati foliations on compact surfaces.
This thesis is concerned with equidistant foliations of Euclidean space, i.e. partitions into complete, connected, properly embedded smooth submanifolds. The space of leaves is an Alexandrov space of nonnegative curvature and the canonical projection is a submetry. Generalizing a result of Gromoll and Walschap we show …
In the present paper we study geometric structures associated with webs of hypersurfaces. We prove that with any geodesic (n+2)-web on an n-dimensional manifold there is naturally associated a unique projective structure and, provided that one of web foliations is pointed, there is also associated a unique affine struc…
Study reveals flatness of Hessian metrics with non-negative Ricci curvature on foliation leaves.
problem Rigidity of Ricci curvature on Hessian manifold leaves.
method Analysis of Ricci curvature properties of Hessian metrics on foliation leaves.
result Non-negative Ricci curvature on a single leaf forces the Hessian metric to be flat and yields bounds on the first Betti number.
Study of affine transformations on topological manifolds, focusing on local freeness and solvability.
problem Understanding the action of affine transformations on topological manifolds.
method Analyzing the subgroup of homeomorphisms that lift to affine transformations and studying the resulting foliation.
result The connected component of the subgroup acts locally freely and is solvable, with additional properties for polynomial manifolds.
Obstruction theory for complex bigraded differential algebras.
problem Understanding extensions and minimal models of bigraded differential algebras with twisted coefficients.
method Development of obstruction theory for Hirsch extensions.
result Proof of uniqueness of relative minimal models and characterization of formality.
We study affine Jacobi structures on an affine bundle π:A→M, i.e. Jacobi brackets that close on affine functions. We prove that there is a one-to-one correspondence between affine Jacobi structures on A and Lie algebroid structures on the vector bundle A+=⋃p∈MAff(Ap,R) of affine functionals. Som…
New bounds on asymptotic dimension for elementary amenable groups.
problem Bounding the asymptotic dimension of box spaces for elementary amenable groups.
method Using subadditivity of asymptotic dimension in group extensions and properties of box spaces.
result For elementary amenable groups, the asymptotic dimension of box spaces is bounded by their Hirsch length, with equality for a specific subclass.
Integral formulas for metric-affine spaces with specific distributions.
problem Finding geometrical obstructions for distributions and foliations.
method Integral formulas involving Ricci and scalar curvatures, second fundamental forms, and integrability tensors.
result Splitting of manifolds and geometrical obstructions for distributions and foliations.
Let (M,w,L) be a symplectic manifold endowed with a lagrangian foliation L. Liberman and Weinstein have shown that the leaves of L are endowed with an affine structure. In this paper we provide links between the theories of affine manifolds and symplectic geometry. Using the work of Donaldson who have shown the existen…
Study embeddings of 3-manifolds in S4 with nilpotent fundamental groups.
problem Embeddings of 3-manifolds in S4 with specific properties of fundamental groups. method Analyzes embeddings with nilpotent fundamental groups, determines groups with Hirsch length ≤5.
result Identifies all nilpotent groups with Hirsch length ≤5 and torsion-free.
The paper explores flat affine and symplectic structures on Lie groups.
problem Exploring flat affine and symplectic structures on Lie groups.
method Left invariant affine structures, immersion of Lie groups, Koszul's method, Lagrangian bi-foliation.
result Flat left invariant affine symplectic connections and their associated affine symplectomorphisms.
A Sim(n-1,1) affine manifold is an affine manifold whose linear holonomy is contained in the similarity lorentzian group but not in the lorentzian group. The class of similarity lorentzian affine manifolds is a small part in the nice class of conformally lorentzian flat manifolds. In this paper we show that a compact S…
Study on characteristic classes for codimension-one foliations, showing non-triviality under specific conditions.
problem Characterizing non-trivial characteristic classes for codimension-one foliations.
method Using Gelfand-Fuchs cohomology and dynamical properties of holonomy groups.
result Non-triviality of certain characteristic classes for Reeb foliations, contradicting classical theorems.
Paper studies symmetries in singular foliations using Lie ∞-morphisms.
problem Understanding symmetries in singular foliations.
method Analyzes Lie ∞-morphisms induced by Lie algebra actions on singular foliations. result Deduces geometrical consequences, including examples of non-extendable symmetries.
Affine manifolds linked to integrable equations and geometric structures.
problem Understanding the geometric and algebraic properties of affine manifolds.
method Analyzing the Kahlerian tangent bundle and multi-dimensional consistency of the TED equation.
result Affine manifolds are related to self-dual Einstein spaces and Hessian structures.
The abstract discusses cohomology of complex manifolds and blow-ups.
problem Calculating cohomologies of complex manifolds and their blow-ups.
method Using sheaf theory and Künneth and Leray-Hirsch theorems.
result Blow-up formulae for complex manifolds are derived.
This paper classifies special ends of real projective orbifolds.
problem Understanding the structures of ends of real projective orbifolds.
method Theoretical tools from affine manifolds and Riemannian foliations.
result Convex but non-properly convex and non-complete-affine radial ends are quasi-joins of horospheres and totally geodesic radial ends.
Let (M,ω) be a symplectic manifold endowed with a agrangian foliation L, it has been shown by Weinstein [16] hat the symplectic structure of M defines on each leaf of L, connection which curvature and torsion forms vanish identically. uppose that L0 is a compact leaf which Weinstein connection …
Affine deformations of convex cones yield special spacetime structures.
problem Deforming divisible convex cones in affine spaces.
method Analyzing the maximal convex domains and quotient structures.
result Quotients of affine actions are MGHCC affine spacetimes.