Contact projective structures have been profoundly studied by D.J.F. Fox. He associated to a contact projective structure a canonical projective structure on the same manifold. We interpret Fox' construction in terms of the equivalent parabolic (Cartan) geometries, showing that it is an analog of Fefferman's constructi…
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48 results for “contact projective geometry”
Study the cuspidal edge's contact with planes and lines in 3D geometry.
problem Classify and understand the singularities of the cuspidal edge's contact with planes and lines.
method Classify submersions on a model of the cuspidal edge by diffeomorphisms, and use singularities of height functions and orthogonal projections to recover and describe the contact.
result Obtained generic singularities and deformations of the apparent contour, related height function and projection singularities to geometric invariants.
Notes on projective, contact, and null curves in complex geometry.
problem Classifying rational null curves in Q^3 of low degree.
method Algebraic geometry and classical background.
result Explicit classification of rational null curves of low degree in Q^3.
A contact projective structure is a contact path geometry the paths of which are among the geodesics of some affine connection. In the manner of T.Y. Thomas there is associated to each contact projective structure an ambient affine connection on a symplectic manifold with one-dimensional fibers over the contact manifol…
Classifies differential operators on symplectic spinors in contact projective geometry.
problem Classifying differential operators on symplectic spinors.
method Classification of homomorphisms of generalized Verma modules and equivariant differential operators.
result Complete classification and construction of differential operators.
Simpler method derived for path geometries on surfaces, characterizing projective path geometries.
problem Characterizing projective path geometries on surfaces.
method Solving the equivalence problem of sub-Riemannian geometry of signature (1,1) on a contact 3-manifold.
result Characterization of projective path geometries in terms of their chains.
Study canonical curves and Kropina metrics in Lagrangian contact geometry.
problem Characterize canonical curves and their relationship to Lagrangian contact structures.
method Construct Fefferman-type spaces, analyze chains and null-chains, use Kropina metrics, apply Fermat principle.
result Chains and null-chains in integrable Lagrangian contact structures are geodesics of Kropina metrics.
Generalizes surgery techniques for projectively Anosov flows.
problem Creating new projectively Anosov flows from existing ones.
method Introducing a generalized Goodman surgery technique for projectively Anosov flows.
result Generates new examples of projectively Anosov flows on hyperbolic 3-manifolds.
Study non-commutative function algebras using contact geometry.
problem Quantize non-commutative function algebras in several variables.
method Contact geometry and rational differential operators.
result Generalizes known constructions of classical equivariants.
The study connects surface geometry in 5D to 4D projections and umbilic curvatures.
problem Understanding the geometry of surfaces in 5D space.
method Relating surfaces in 5D to surfaces in 4D via projections and normal sections, analyzing asymptotic directions and umbilic curvatures.
result Relations between asymptotic directions and umbilic curvatures in 5D surfaces and their counterparts in 4D projections.
Geometry of third-order ODEsmath.DG
We address the problem of local geometry of third order ODEs modulo contact, point and fibre-preserving transformations of variables. Several new and already known geometries are described in a uniform manner by the Cartan method of equivalence. This includes conformal, Weyl and metric geometries in three and six dimen…
Bi-contact surgery operations can be applied to Anosov flows.
problem Characterizing Anosov flows and their properties.
method Metric and contact geometric characterizations, Liouville geometry, Reeb dynamics.
result Bi-contact surgery operations can be applied to Anosov flows.
Abstract: Study of surface transitions and IDE inflections via contact geometry.
problem Understanding transitions on surfaces and implicit differential equations.
method Contact geometry and Legendrian properties of projections.
result List of unavoidable local phenomena on surfaces and IDE solutions.
Study on parabolic points and cylindrical surfaces in Euclidean 3-space.
problem Characterizing parabolic points and their geometric properties.
method Introducing contact cylindrical surfaces and analyzing their properties.
result Characterization of -singularity through projections.
The study explores autonomous systems and their connections to contact geometry and Frobenius manifolds.
problem Understanding the connections between autonomous systems and geometric structures.
method Investigation of the Darboux-Halphen-Ramanujan system, contact geometry, and Frobenius manifolds.
result Highlighting the role of contact geometry in autonomous systems.
We give a classification of order invariant differential operators acting between sections of certain bundles associated to Cartan geometries of the so called metaplectic contact projective type. These bundles are associated via representations, which are derived from the so called higher symplectic, harmonic …
Complex contact manifolds, varieties of minimal rational tangents, and exterior differential systemsmath.AG
The abstract discusses classification theorems for complex contact manifolds and their geometric properties.
problem Classifying complex contact manifolds and understanding their geometric properties.
method Analyzing contact lines and varieties of minimal rational tangents.
result Partial classification theorems for projective complex contact manifolds.
Develops a universal Hermitian projective calculus for complex hyperbolic two-space
problem Complex hyperbolic geometry
method Algebraic invariant calculus
result Denominator-cleared identities for various geometric quantities
Study on structures and almost para-contact structures in 7D.
problem Understanding the relation between structures and almost para-contact structures.
method Calculating projections using properties of structures.
result Determined the class of almost para-contact structures induced by structures.
Study on generalized quasi-Einstein structures in contact geometry.
problem Characterizing and understanding generalized quasi-Einstein structures in contact geometry.
method Investigation of properties, existence, and characterizations of generalized quasi-Einstein normal metric contact pair manifolds.
result Normal metric contact pair manifolds with generalized quasi-constant curvature are generalized quasi-Einstein manifolds.
Geometry of Third-Order Ordinary Differential Equations and Its Applications in General Relativitymath.DG
A PhD thesis written under supervision of Pawel Nurowski and defended at the Faculty of Physics of the University of Warsaw. We adress the problems of local equivalence and geometry of third order ODEs modulo contact, point and fibre-preserving transformations of variables. Several new and already known geometries are …
3-Sasaki structures linked to projective geometry.
problem Understanding 3-Sasaki structures via projective geometry.
method Establishing a connection between 3-Sasaki structures and projective structures with specific holonomy reductions.
result 3-Sasaki structures are described as projective structures with a particular holonomy reduction to the unitary quaternionic group.
Investigate the local geometry of smooth surfaces in 4-space via contact with 2-planes and apparent contours.
problem Local geometry of smooth surfaces in 4-space
method Contact with 2-planes and apparent contour
result Prove connections between singularities of parallel projections, orthogonal projections, and height functions.
The paper studies reducibility properties in Sasakian geometry, classifying certain contact structures and extremal metrics.
problem Reducibility properties in Sasakian geometry, focusing on contact structures and extremal metrics.
method Developed the Sasaki version of the de Rham Decomposition Theorem, introduced cone reducible concept, and classified Sasakian structures.
result Classified all Sasakian structures up to contact isotopy on bundles over a Riemann surface of genus greater than zero, and showed extremal Sasaki metrics split in the toric case.
Study aerodynamics of flying saucers on curved spaces.
problem Understanding the motion of flying saucers on different types of curved surfaces.
method Identifying structures on the configuration space of flying saucers and relating them to the geometry of the underlying curved manifold.
result The symmetries of the flying saucer's motion can be described by the split form of the exceptional Lie algebra G2 when the manifold has a certain type of structure.
Chains in CR geometry are geodesics of a Kropina metric.
problem Understanding chains in CR geometry.
method Generalization of Fermat principle and Kropina metric construction.
result Chains determine CR structure up to conjugacy.
Study of rational curves in complex manifolds with specific normal bundles.
problem Characterizing rational curves in complex manifolds with given normal bundles.
method Analyzing differential and projective geometric properties of rational curves and their tangents.
result Classification of rational curves into Goursat and Cartan types based on their geometric properties.
Study contact structures on projective spaces, proving infinite non-isotopic structures.
problem Classify contact structures on projective spaces with specific surgery numbers.
method Detailed analysis of Gompf's Γ-invariant and d_3-invariant of tangential 2-plane fields.
result Infinitely many non-isotopic contact structures on real projective 3-space.
Projective geometry simplifies Sasaki-Einstein structures and their compactification.
problem Understanding and compactifying Sasaki-Einstein structures.
method Projective differential geometry and holonomy reductions.
result Characterization of Sasaki-Einstein structures and their compactification.
Let X be a complex Fano-manifolds with second Betti-number 1 which carries a contact structure. It follows from previous work that such a manifold can always be covered by lines. Thus, it seems natural to consider the geometry of lines in greater detail. In this brief note we show that if x in X is a general point, the…
New construction shows VMRTs of unbendable curves can be Legendrian.
problem Characterize VMRTs of unbendable rational curves under contact structures.
method Used geometry of contact lines and symplectic geometry of distributions.
result VMRTs of Legendrian submanifolds can be realized.
We study equivariant contact structures on complex projective varieties arising as partial flag varieties , where is a connected, simply-connected complex simple group of type and is a parabolic subgroup. We prove a special case of the LeBrun-Salamon conjecture for partial flag varieties of these typ…
H. Sato introduced a Schwarzian derivative of a contactomorphism of three-dimensional Euclidean space and with T. Ozawa described its basic properties. In this note their construction is extended to all odd dimensions and to non-flat contact projective structures. The contact projective Schwarzian derivative of a conta…
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 consider a contact manifold with a pseudo-Riemannian metric and define a contact vector field intrinsically associated to this pair of structures. We call this new differential invariant the contact Riemannian curl. On a Riemannian manifold, Killing vector fields are those that annihilate the metric; a Killing -f…
A long-standing open problem in systolic geometry asks whether a Riemannian metric on the real projective space whose volume equals that of the canonical metric, but is not isometric to it, must necessarily carry a periodic geodesic of length smaller than π. A contact-geometric reformulation of systolic geometry and th…
The aim of this paper and its sequel is to introduce and classify the holonomy algebras of the projective Tractor connection. After a brief historical background, this paper presents and analyses the projective Cartan and Tractor connections, the various structures they can preserve, and their geometric interpretations…
We discuss contact invariant structures on the space of solutions of a third-order ordinary differential equation. Associated to any third-order differential equation modulo contact transformations, Chern introduced a degenerate conformal Lorentzian metric on the space of 2-jets of functions of one variable. When the W…
Study para-CR structures in 5D with degenerate Levi form, revealing geometric conditions for conic graphs and Lorentzian ODEs.
problem Investigate invariant properties of para-CR structures in 5D with degenerate Levi form.
method Analyze basic invariants and their vanishing conditions to establish geometric interpretations and necessary conditions.
result Vanishing of the third basic invariant implies contact projective geometries on quotient spaces.
We investigate certain natural connections between subriemannian geometry and hyperbolic dynamical systems. In particular, we study dynamically defined horizontal distributions which split into two integrable ones and ask: how is the energy of a subriemannian geodesic shared between its projections onto the integrable …
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.
Classifies and analyzes the stability of black hole event horizon birth points using contact geometry.
problem Classifying and understanding the structural possibilities of black hole crease sets.
method Contact geometry approach, focusing on BigFronts and their Legendrian projections.
result Refined stability discussion of the event horizon birth component and identification of additional components.
Abstract: Survey on contact submanifolds.
problem Understanding contact submanifolds.
method None specified, survey of existing knowledge.
result Discussion of various contact submanifolds.
Study of symplectic Monge-Ampère equations using moment maps and contact structures.
problem Characterizing symplectic Monge-Ampère equations through geometric structures.
method Constructing contact cone structures and using moment maps to relate equations to projective spaces.
result The contact cone structure and the cocharacteristic variety coincide for non-degenerate equations.
Spinors help study unique five-dimensional contact structures.
problem Understanding unique five-dimensional contact structures.
method Using classical two-component spinors, define directional derivatives in contact directions.
result Calculate invariant torsion of contact structure.
Discusses the tight versus overtwisted dichotomy in 3D contact geometry.
problem Understanding the tight versus overtwisted dichotomy in 3D contact geometry.
method Reviews Eliashberg's seminal work and contributions to the theory.
result Explains the genesis and importance of the tight versus overtwisted dichotomy.
Infinite-dimensional contact geometry explored.
problem Generalizing contact geometry to infinite dimensions.
method Generalization of cosymplectic, contact, and cocontact manifolds to infinite dimensions.
result Model examples of time-dependent and dissipative Hamiltonian systems calculated.
A car as parabolic geometrymath.DG
A car's motion shows flat parabolic geometry.
problem Understanding the geometry of a car's motion.
method Analyzing a car as a nonholonomic system.
result Shows a car's motion as a flat parabolic geometry.