We develop a construction of Engel stuctures on 4-manifolds based on decompositions of manifolds into round handles. This allows us to show that all parallelizable 4-manifolds admit an Engel structure. We also show that, given two Engel manifolds M_1,M_2 satisfying a certain condition on the characteristic foliation, t…
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48 results for “Engel relation”
We give two applications of the 2-Engel relation, classically studied in finite and Lie groups, to the 4-dimensional topological surgery conjecture. The A-B slice problem, a reformulation of the surgery conjecture for free groups, is shown to admit a homotopy solution. We also exhibit a new collection of universal surg…
We describe a complete system of invariants for 4-dimensional CR manifolds of CR dimension 1 and codimension 2 with Engel CR distribution by constructing an explicit canonical Cartan connection. We also investigate the relation between the Cartan connection and the normal form of the defining equation of an embedded En…
The paper explores geometric properties of Engel structures on 4-manifolds.
problem Geometric and Riemannian properties of Engel structures on 4-manifolds.
method Study of Engel defining forms, Reeb distribution integrability, and vector fields preserving the structure.
result Conditions for integrability of the Reeb distribution and construction of Engel defining forms.
Solved A-B slice problem using link-homotopy+.
problem 4D topological surgery conjecture for free groups.
method Geometric applications of 2-Engel relation.
result Solved A-B slice problem.
Paper studies Engel structures and automorphisms on 4-manifolds.
problem Understanding Engel structures and their automorphisms.
method Developing maps and Cartan prolongations of contact 3-orbifolds.
result Automorphism groups of Engel manifolds are embedded into automorphism groups of contact 3-orbifolds.
Engel knots map to formal knots without restrictions.
problem Classifying Engel knots and their properties.
method Weak homotopy equivalence of Engel knots to formal knots.
result Engel knots map to formal knots without restrictions.
Engel structures on complex surfaces are classified based on Chern classes.
problem Classifying complex surfaces with specific Engel structures.
method Using Chern classes and adapting Geiges' construction.
result Engel structures on complex surfaces with trivial Chern classes exist and are unique.
Modified Engel structures allow complete h-principle for overtwisted discs.
problem Engel structures and their overtwisted discs.
method Engel twist modification and h-principle proof.
result Complete h-principle for overtwisted Engel structures.
Lagrangian Engel Structuresmath.DG
Study of Lagrangian Engel structures on symplectic 4-manifolds.
problem Understanding the geometry of Engel structures on symplectic 4-manifolds.
method Solving equivalence problems and classifying homogeneous examples.
result Determine all compact, homogeneous examples of Lagrangian Engel structures.
The study examines the stability of Engel-like structures in higher dimensions.
problem Stability of Engel-like distributions in higher dimensions.
method Motivated by Cartan prolongation of contact manifolds, the article introduces a higher-dimensional analogue of Engel structures and investigates their stability.
result Generalizes Gray-type stability to Engel manifolds in higher dimensions.
Complex Engel Structuresmath.DG
Study complex Engel structures on complex surfaces, classifying homogeneous examples.
problem Classify homogeneous complex Engel structures on complex surfaces.
method Solve equivalence problems, use structure equations, classify examples.
result Determine all compact, homogeneous complex Engel structures.
We call two Engel structures isotopic if they are homotopic through Engel structures by a homotopy that fixes the characteristic line field. In the present paper we define an isotopy invariant of Engel structures on oriented circle bundles over closed oriented three-manifolds and apply it to give an isotopy classificat…
Researchers create non-isomorphic holomorphic Engel structures on C4.
problem Constructing non-isomorphic holomorphic Engel structures on C4.
method Controlled curves and distributions to create Engel structures.
result Existence of uncountably many non-isomorphic holomorphic Engel structures on C4.
Loose Engel structuresmath.SG
The article classifies Engel structures up to homotopy.
problem Classifying Engel structures up to homotopy.
method Introduced the notion of a loose family of Engel structures and showed homotopy equivalence conditions.
result Two Engel structures are homotopic if and only if they are formally homotopic.
New surgery problems solved for 4D topological surgery.
problem Intermediate 4D surgery problems for 1/2--null groups.
method Group-theoretic 2-Engel relation and geometric applications.
result 1/2--null surgery problems are universal.
The study classifies prolongations up to Engel homotopy based on their formal data.
problem Classifying prolongations up to Engel homotopy.
method Reduction to formal data and study of homotopy type.
result The classification problem reduces to formal data when the turning number is large enough.
Modeling curvature-sensitive cells in visual cortex with geometric structures.
problem Understanding the functional architecture of curvature-sensitive cells in the visual cortex.
method Geometric model based on Engel structure and SIM(2) symmetry.
result Identified SIM(2) as the natural symmetry group for curvature-sensitive cells.
Study of marked contact Engel structures with geometric invariants.
problem Understanding the geometry of marked contact Engel structures.
method Investigation of local geometry and classification of homogeneous models.
result Proved an analogue of the Kerr theorem for marked contact Engel structures.
Study uses superalgebra homology to classify 4D Engel-like Lie algebras.
problem Classifying 4-dimensional Engel-like Lie algebras.
method Applied homology groups of Lie superalgebras.
result Distinguished and classified 4D Engel-like Lie algebras.
An Engel structure is a maximally non-integrable field of two-planes tangent to a four-manifold. Any two such structures are locally diffeomorphic. We investigate the space of global deformations of canonical Engel structures arising out of contact three-manifolds. The main tool is Cartan's method of prolongation and d…
Engel structures and flows on 4-manifolds linked via constant rank intersections.
problem Pairs of Engel structures on 4-manifolds with constant rank intersections.
method Established a correspondence with weakly hyperbolic flows.
result Engel structures and flows on 4-manifolds are linked via constant rank intersections.
Expanding knowledge of Engel structures via geometric constructions and dynamics analysis.
problem Understanding Engel structures and their geometric properties.
method Developing canonical geometric constructions and analyzing dynamics of Cauchy characteristics.
result Illustrates the elliptic, parabolic, and hyperbolic behaviors of Cauchy characteristics.
Engel manifolds show transverse tori can be made to have various formal invariants.
problem Understanding transverse tori in Engel manifolds.
method Analogous to transverse knots, classify formal invariants and show their uniqueness.
result Engel manifolds can have infinitely many transverse isotopy classes of tori with specific invariants.
Study finds non-isotopic transverse tori in Engel manifolds.
problem Identifying distinct transverse tori in Engel manifolds.
method Constructing an infinite family of non-isotopic transverse tori, introducing a homological invariant.
result Found an infinite family of non-isotopic transverse tori that are smoothly isotopic.
New integrable structures found with abnormal geodesics.
problem Integrable homogeneous sub-Riemannian structures with abnormal geodesics.
method Analysis of equivalence problem for sub-Riemannian Engel structures.
result First known family of examples of integrable homogeneous sub-Riemannian structures with strictly abnormal geodesics.
There is a remarkable type of field of two-planes special to four dimensions known as an Engel distributions. They are the only stable regular distributions besides the contact, quasi-contact and line fields. If an arbitrary two-plane field on a four-manifold is slightly perturbed then it will be Engel at generic point…
Engel CR submanifolds of math.DG
Engel structures on bundles over 3-manifolds in complex 3-space.
problem Embedding bundles over 3-manifolds into complex 3-space with Engel structures.
method Sufficient condition for -bundles to admit immersions/embeddings with complex tangencies defining Engel structures.
result Every oriented -bundle over a closed, oriented 3-manifold admits an immersion with complex tangencies defining Engel structures.
Modeling curvature-sensitive cells in visual cortex using manifold geometry.
problem Understanding how curvature influences cell function in the visual cortex.
method Developed a 4D manifold with canonical Engel structure to represent orientation, position, curvature, and scale.
result Characterized curvature-sensitive receptive profiles using left-invariant generators of the Engel structure.
New 4D shapes can't be opened like books.
problem Existence of 4D shapes without open book decompositions.
method Demonstrated existence of infinitely many parallelizable 4-manifolds without open book decompositions.
result No open book decomposition for certain Engel manifolds.
The paper studies extremals in the Engel group with a sub-Lorentzian metric.
problem Optimizing trajectories in the Engel group with a specific metric.
method Proves local maximality of timelike normal extremals, parametrizes trajectories, describes fixed points, and estimates cut times.
result Timelike normal extremals are locally maximizing and have specific parametrizations and fixed points.
Classifies metric lines in Engel-type groups, a step towards solving sub-Riemannian manifold problems.
problem Classifying metric lines in Engel-type groups.
method Sequence method to study metric lines in jet space.
result Classified metric lines of Engel-type groups $\Eng(n)$.
A simple proof is given of the following result first observed by J. Adachi: embedded circles tangent to the standard Engel structure on Euclidean 4-space are classified, up to isotopy via such embeddings, by their rotation number.
Researchers found optimal paths on a specific geometric group.
problem Finding optimal paths in a geometric group with a sub-Finsler metric.
method Used Pontryagin Maximum Principle for time-optimal control problem.
result Found extremals for left-invariant sub-Finsler metric.
In part I it was shown that for each k>0 the generalized Sato-Levine invariant detects a gap between k-quasi-isotopy of link and peripheral structure preserving isomorphism of the finest quotient G_k of its fundamental group, `functorially' invariant under k-quasi-isotopy. Here we show that Cochran's derived invariant …
We apply spectral sequences to derive both an obstruction to the existence of -fold prolongations and a topological classification. Prolongations have been used in the literature in an attempt to prove that every Engel structure on with characteristic line field tangent to the fibers is determi…
On the space of null geodesics of a spacetime: the compact case, Engel geometry and retrievabilitymath.DG
Study null geodesics on specific spacetimes, finding contact manifolds and Engel geometry applications.
problem Computing the space of null geodesics for a family of spacetimes.
method Computed the contact manifold of null geodesics for a specific family of spacetimes using Engel geometry.
result Characterized the contact manifolds of null geodesics and retrieved the spacetime.
We find all intrinsic measures of smooth submanifolds in the Engel group, showing that they are equivalent to the corresponding -dimensional spherical Hausdorff measure restricted to the submanifold. The integer is the degree of the submanifold. These results follow from a different approach to negligi…
Dancing polygons and rolling balls linked via a special geometric distribution.
problem Understanding the geometric and mechanical relationship between dancing polygons and rolling balls.
method Mapping dancing polygons to trajectories of a rolling ball on a 3D surface, both described by a specific geometric distribution.
result Non-degenerate dancing pairs of polygons exist for all and correspond to rolling ball trajectories.
We construct normal forms for Lorentzian metrics on Engel distributions under the assumption that abnormal curves are timelike future directed Hamiltonian geodesics. Then we indicate some cases in which the abnormal timelike future directed curve initiating at the origin is geometrically optimal. We also give certain e…
The study finds that the export shares of machinery and food/crude materials are significantly correlated with GDP.
problem Understanding the relationship between export shares and GDP across different commodity sectors.
method Analysis of GDP and international trade data using the SITC classification from 1962 to 2000.
result The export shares of machinery and food/crude materials are significantly correlated with GDP, following a power-law relationship.
Alternative construction of Rumin complex on Lie groups.
problem Constructing Rumin complex on homogeneous nilpotent Lie groups.
method Using ideas from parabolic geometry, an alternative construction to the classical one on Carnot groups.
result Explicit computations for the Engel group using the new approach.
Study curvature in Carnot groups with rank-two distributions.
problem Understanding curvature in specific geometric structures.
method Generalized sectional curvature for Carnot groups with rank-two distributions.
result Curvature depends on the Engel part in certain Carnot groups.
This paper extends geometric structure theory to infinite type structures.
problem Calculating characteristic class relations in complex Cartan geometries.
method Improves representation theory for infinite type structures.
result Direct calculation of characteristic class relations from structure group representation.
Engel structures on M x S^1 and M x I are studied in this paper, where M is a 3-dimensional manifold. We suppose that these structures have characteristic line fields parallel to the fibres, S^1 or I. It is proved that they are characterized by contact structures on the cross section M, the twisting numbers, and Legend…
We realise the first and second Grushin distributions as symmetry reductions of the 3-dimensional Heisenberg distribution and 4-dimensional Engel distribution respectively. Similarly, we realise the Martinet distribution as an alternative symmetry reduction of the Engel distribution. These reductions allow us to derive…
Classification of homogeneous almost complex 4-manifolds with non-degenerate torsion bundle
problem Classification of homogeneous almost complex 4-manifolds with non-degenerate torsion bundle
method Classification using rigidity of structures
result Complete classification in the homogeneous setting
This paper characterizes semigenerated Carnot groups and applies it to rectifiability of perimeter sets.
problem Characterizing semigenerated Carnot groups and their applications to rectifiability.
method Algebraic approach focusing on semigroup generation and Engel-type quotients.
result Complete characterization of semigeneration in Carnot groups of step 3 and sufficient criteria for semigeneration in Carnot groups of arbitrary step.