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.
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.
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…
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.
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.
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.
We study pairs of Engel structures on four-manifolds whose intersection has constant rank one and which define the same even contact structure, but induce different orientations on it. We establish a correspondence between such pairs of Engel structures and a class of weakly hyperbolic flows. This correspondence is ana…
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.
We apply spectral sequences to derive both an obstruction to the existence of n-fold prolongations and a topological classification. Prolongations have been used in the literature in an attempt to prove that every Engel structure on M×S1 with characteristic line field tangent to the fibers is determi…
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…
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.
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.
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 give local descriptions of parabolic contact structures and show how their flat models yield explicit PDE having symmetry algebras isomorphic to all complex simple Lie algebras except sl2. This yields a remarkably uniform generalization of the Cartan-Engel models from 1893 in the G2 case. We give a …
Researchers compute contact structures for null geodesics on specific spacetimes.
problem Understanding the canonical contact structure of null geodesics in spacetimes.
method Explicit calculations for specific spacetimes, including lens spaces and three-dimensional spacetimes.
result Contact structures on null geodesics are derived from the Lorentz prolongation of spacetimes.
Study magnetic flows on 3D contact sub-Riemannian manifolds using Rumin complex.
problem Understanding magnetic flows on 3D contact sub-Riemannian manifolds.
method Introducing horizontal magnetic flows via closed Rumin differential two-forms and analyzing the lifted sub-Riemannian structure.
result Horizontal magnetic flows can be interpreted as geodesic flows on a suitably lifted structure, which is of Engel type when the magnetic field is non-vanishing.
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.
Notes on sub-Riemannian geometry equivalence problem.
problem Isometric sub-Riemannian manifolds equivalence.
method Introduction to connections, frame bundles, and sub-Riemannian geometry; description of canonical grading and connection.
result Minimal set of isometries for Engel (2,3,4)-manifolds.
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.
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.
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 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…
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.
Develops k-contact geometry theory for field theories.
problem Analyse field theories using k-contact geometry.
method Distributions maximally non-integrable with k commuting Lie symmetries.
result Established k-contact distributions and their relationships.
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.
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.
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.
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.
We provide the first known family of examples of integrable homogeneous sub-Riemannian structures admitting strictly abnormal geodesics. These examples were obtained through the analysis of the equivalence problem for sub-Riemannian Engel structures. We formulate a criterion of strict abnormality in terms of structure …
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 S1-bundles to admit immersions/embeddings with complex tangencies defining Engel structures. result Every oriented S1-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.
A flying saucer's flight is mathematically modeled with geometric structures.
problem Modeling the complex flight maneuvers of a flying saucer.
method Imposing nonlinear restrictions on the saucer's velocity to define geometric structures.
result Three types of flat parabolic geometries are defined for the saucer's configuration space.
The paper extends local h-principles to complex structures on Stein manifolds.
problem Existence of local h-principles for complex structures on Stein manifolds.
method Introducing realifications of partial holomorphic relations and proving h-principles for them.
result Local h-principles can be extended to complex structures on Stein manifolds.
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.
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.
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 $\Eng(n)$.
The Engel group's sub-Riemannian structure is analyzed, revealing unique geometric features.
problem Analyzing the sub-Riemannian structure on the Engel group.
method Global structure of the cut locus, Maxwell set, and caustic described.
result Cut locus has 6 three-dimensional strata, 12 two-dimensional strata, and 2 one-dimensional strata.
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 notion of curvature discussed in this paper is a far going generalization of the Riemannian sectional curvature. It was first introduced by Agrachev, Barilari and Rizzi in arXiv:1306.5318, and it is defined for a wide class of optimal control problems: a unified framework including geometric structures such as Riem…
The left-invariant sub-Riemannian problem on the Engel group is considered. The problem gives the nilpotent approximation to generic nonholonomic systems in four-dimensional space with two-dimensional control, for instance to a system which describes motion of mobile robot with a trailer. The global optimality of extre…
Reduced sub-Riemannian time on a specific group structure.
problem Optimizing paths in a sub-Riemannian structure on a Carnot group.
method Proved conjectured cut times, compared with known results, and solved equations in elliptic functions.
result Reduced cut times for sub-Riemannian paths on the Cartan group.
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.
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…
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.
A Goursat structure on a manifold of dimension n is a rank two distribution D such that dim D(i)=i+2, for i=0,...,n-2, where D(i) denotes the derived flag of D, which is defined by D(0)=D and D(i+1)=D(i)+[D(i),D(i)]. Goursat structures appeared first in the work of E. von Weber and E. Cartan, who have shown that on an …
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
We find all intrinsic measures of C1,1 smooth submanifolds in the Engel group, showing that they are equivalent to the corresponding d-dimensional spherical Hausdorff measure restricted to the submanifold. The integer d is the degree of the submanifold. These results follow from a different approach to negligi…