Study symplectic and Hamiltonian actions on irrational ruled surfaces, proving existence and non-existence of extensions.
problem Tackles the extension of symplectic and Hamiltonian cyclic actions to Hamiltonian circle actions on irrational ruled symplectic 4-manifolds.
method Constructs symplectic involutions and cyclic actions, classifies symplectic morphisms, and proves non-extendability of certain actions.
result Shows existence and non-existence of Hamiltonian circle actions for different cyclic actions on irrational ruled symplectic 4-manifolds.
Study irrational rotations and construct 2-filling rays on infinite type surfaces.
problem Understanding dense orbits in skew product transformations.
method Skew product transformation with continued fraction analysis.
result Existence of infinite cliques of 2-filling rays on infinite type surfaces.
We describe Veech groups of flat surfaces arising from irrational angled polygonal billiards or irreducible stable abelian differentials. For irrational polygonal billiards, we prove that these groups are non-discrete subgroups of SO(2,R) and we calculate their rank.
New coefficient detects irrational rotation behavior on infinite-type surfaces.
problem Detecting irrational rotation behavior on surfaces of infinite type.
method Introducing a new quasimorphism, the Dehn twist coefficient, and proving its properties.
result The Dehn twist coefficient can have image all of R for some infinite-type surfaces.
The paper solves isotopy problems on 4-manifolds and classifies symplectic structures.
problem Isotopy problems on 4-manifolds and uniqueness of symplectic structures.
method Generalization of Dax invariant to embedded closed surfaces, symplectic topology techniques.
result Infinitely many non-isotopic symplectic forms on irrational ruled surfaces.
Study irrational pencils on complex manifolds, finding non-finitely generated homology.
problem Understanding the homology of the kernel induced by irrational pencils on complex manifolds.
method Analyzing critical points and homology of fundamental groups of complex manifolds.
result Homology of the kernel of the morphism induced by the pencil on fundamental groups is not finitely generated.
In this paper, the symplectic genus for any 2-dimensional class in a 4-manifold admitting a symplectic structure is introduced, and its relation with the minimal genus is studied. It is used to describe which classes in rational and irrational ruled manifolds are realized by connected symplectic surfaces. In particular…
We consider the sub-Riemannian metric gh on S3 provided by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the Carnot-Carathéodory distance and we show that, depending on their curvature, they …
Study tiling spaces over irrational tori using diffeological classification.
problem Understanding the structure of tiling spaces over irrational tori.
method Diffeological classification of irrational tori and analysis of fiber bundle structures.
result Inherited diffeological equivalence of one-dimensional tiling spaces over irrational tori.
Following a Geometrical Brownian Motion extension into an Irrational Fractional Brownian Motion model, we re-examine agent behaviour reacting to time dependent news on the log-returns thereby modifying a financial market evolution. We specifically discuss the role of financial news or economic information positive or n…
We show that the Novikov-Shubin invariant of an element of the integral group ring of the lamplighter group Z_2 \wr Z can be irrational. This disproves a conjecture of Lott and Lueck. Furthermore we show that every positive real number is equal to the Novikov-Shubin invariant of some element of the real group ring of Z…
Symplectic embeddings of balls into specific manifolds are studied, with restrictions and obstructions identified.
problem Understanding symplectic embeddings of balls into complex projective spaces, tori, and K3 surfaces.
method Analyzing embeddings with respect to complex structures compatible with the symplectic form and identifying obstructions.
result Symplectic volume is the primary obstruction for the existence of embeddings of balls into certain manifolds.
We prove that there are examples of finitely generated groups G together with group ring elements Q \in \bbQ G for which the von Neumann dimension \dim_{LG}\ker Q is irrational, so (in conjunction with other known results) answering a question of Atiyah.
In this study, we give the relationships between the conical curvatures of ruled surfaces drawn by the unit vectors of the ruling, central normal and central tangent of a regular ruled surface in the Euclidean -space. We obtain the differential equations characterizing slant ruled surfaces and if the reference ruled su…
In this study, we define some new types of ruled surfaces called slant ruled surfaces. We give some characterizations for a regular ruled surface to be a slant ruled surface in Euclidean 3- space. We show that if the slant ruled surface is developable then the striction curve is a general helix or a slant helix accordi…
Study ruled surfaces with finite multiplicity, focusing on their curves and singularities.
problem Understanding ruled surfaces with finite multiplicity.
method Analyzing striction curves and singularities of ruled surfaces.
result Geometric meanings of invariants related to ruled surfaces.
We study the structure of the stable norm of Finsler metrics on the 2-torus with a focus to points of irrational slope. By our results, the stable norm detects KAM-tori and hyperbolicity in the geodesic flow. Moreover, we study the stable norm in some natural examples.
We give examples of finitely presented groups containing elements with irrational (in fact, transcendental) stable commutator length, thus answering in the negative a question of M. Gromov. Our examples come from 1-dimensional dynamics, and are related to the generalized Thompson groups studied by M. Stein, I. Liousse …
In this study, we introduce Darboux slant ruled surfaces in the Euclidean 3-space which is defined by the property that the Darboux vector of orthonormel frame of ruled surface makes a constant angle with a fixed, non-zero direction. We obtain the characterizations of Darboux slant ruled surfaces regarding the conical …
In this study, we define some new types of non-null ruled surfaces called slant ruled surfaces in the Minkowski 3-space E_1^3. We introduce some characterizations for a non-null ruled surface to be a slant ruled surface in E_1^3. Moreover, we obtain some corollaries which give the relationships between a non-null slant…
In this study, we consider the notion of similar ruled surface for timelike and spacelike ruled surfaces in Minkowski 3-space. We obtain some properties of these special surfaces in E_1^3 and we show that developable ruled surfaces in E_1^3 form a family of similar ruled surfaces if and only if the striction curves of …
In this study, we define a family of ruled surfaces in the Euclidean 3-space E^3 and called similar ruled surfaces. We obtain some properties of these special surfaces and we show that developable ruled surfaces form a family of similar ruled surfaces if and only if the striction curves of the surfaces are similar curv…
Study embeds ruled surfaces into symplectic manifolds, finds Stein fillability results.
problem Embedding rational ruled surfaces into symplectic manifolds.
method Analyzes symplectic hyperplane sections of rational ruled surfaces.
result Obtains Stein fillability results for rational ruled surfaces.
New type of ruled surfaces studied with properties and examples.
problem Characterizing and understanding new types of ruled surfaces.
method Definition of a new orthonormal frame, calculation of Gaussian and mean curvatures, analysis of Weingarten map and geodesic properties.
result Conditions for an OT-surface to be flat or minimal are derived, and examples of helices and slant helices are provided.
In this paper, using the classifications of timelike and spacelike ruled surfaces, we study the Mannheim offsets of timelike ruled surfaces in Minkowski 3-space. Firstly, we define the Mannheim offsets of a timelike ruled surface by considering the Lorentzian casual character of the offset surface. We obtain that the M…
We classify ruled minimal surfaces in R3 with density ez. It is showed that there is no noncylindrical ruled minimal surface and there is a family of cylindrical ruled minimal surfaces in R3 with density ez. It is also proved that all translation minimal surfaces are ruled.
The Lamarle Formula, given by Kruppa in \cite{Kr}, is known as a relationship between the Gaussian curvature and the distribution parameter of a ruled surface in the surface theory. The ruled surfaces were investigated in 3 different classes with respect to the character of base curves and rulings, \cite{Tu1},\cite{Tu2…
In this paper, we define dual geodesic trihedron(dual Darboux frame) of a spacelike ruled surface. Then, we study Mannheim offsets of spacelike ruled surfaces in dual Lorentzian space by considering the E. Study Mapping. We represent spacelike ruled surfaces by dual Lorentzian unit spherical curves and define Mannheim …
Study classifies ruled surfaces critical to Dirichlet energy.
problem Identifying ruled surfaces critical to Dirichlet energy.
method Explicit parametrization of ruled surfaces.
result Classification of ruled surfaces as critical points of Dirichlet energy.
In this paper, we consider non developable ruled surface with spacelike ruling, timelike ruling, respectively. We give the relations between the structure functions with the curvature and torsion of the striction line of the timelike and spacelike non developable ruled surfaces. Also, we have calculated the gaussian an…
Defines new ruled surfaces using alternative frames and analyzes their geometric properties.
problem Analyzing geometric properties of new ruled surfaces.
method Using alternative frames to define new ruled surfaces and studying their differential geometric properties.
result Geodesic curvatures, normal curvatures, and geodesic torsions of the base curve and striction line are found.
In this paper, we introduce the dual geodesic trihedron (dual Darboux frame) of a timelike ruled surface. By the aid of the E. Study Mapping, we consider timelike ruled surfaces as dual hyperbolic spherical curves and define the Mannheim offsets of timelike ruled surfaces by means of dual Darboux frame. We obtain the r…
This study examines geometric properties and offsets of slant timelike-ruled surfaces.
problem Geometric properties and offsets of slant timelike-ruled surfaces in Minkowski 3-space.
method Derivation of parametric formulation, conditions for coaxial alignment, examination through Blaschke and Darboux frames.
result Conditions ensuring the coaxial alignment of the central normal with the ruling direction of the offset surface.
Paper classifies ruled surfaces in Lorentz-Minkowski space for a specific flow.
problem Classifying ruled surfaces in Lorentz-Minkowski space.
method Examining homothetic self-similar solutions of the inverse mean curvature flow.
result Existence of two classes of non-cylindrical homothetic solitons.
The paper classifies ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia.
problem Classifying ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia.
method Maximum principle applications, classification based on causal character of rulings.
result Planes are the only cylindrical stationary surfaces. For non-cylindrical surfaces, classification depends on the causal character of the rulings.
Defines and analyzes generalized normal ruled surfaces of curves in 3D space.
problem Understanding the geometry of generalized normal ruled surfaces.
method Calculates Gaussian and mean curvatures to determine surface properties and examines curve conditions.
result Determines when surfaces are flat or minimal and identifies specific curve types.
Study on Kähler metrics on ruled surfaces, proving existence and non-existence.
problem Existence and non-existence of Kähler metrics on minimal ruled surfaces.
method Analysis of twisted and coupled constant scalar curvature Kähler metrics.
result Bound for Chen-Cheng invariant on ruled surfaces.
In this paper, we obtain the characterizations of Mannheim offsets of the timelike ruled surface with spacelike rulings in dual Lorentzian space. We give the relations between terms of their integral invariants and also we give the new characterization of the Mannheim offsets of developable timelike ruled surface. More…
In this paper, we study the special curves and ruled surfaces on helix hypersurface whose tangent planes make a constant angle with a fixed direction in Euclidean n-space Besides, we observe some special ruled surfaces in and give requirement of being developable of the ruled surface. Also, we investigate the helix sur…
Study infinite symplectic forms on ruled surfaces.
problem Uniqueness of symplectic structures on four-manifolds.
method Provided an infinite family of non-isotopic symplectic forms.
result Symplectic forms on ruled surfaces are pairwise non-isotopic.
Study of timelike surfaces with time-minimizing rulings in Newtonian and relativistic spacetimes.
problem Understanding time-minimizing paths in spacetime geometries.
method Constructing timelike surfaces ruled by geodesics of Finsler or Jacobi metrics.
result Explicit examples of brachistochrone-ruled timelike surfaces in Minkowski and Schwarzschild spacetimes.
Study classifies ruled surfaces from mean curvature flow solutions.
problem Investigating surfaces with specific mean curvature equations.
method Classifying ruled and translation surfaces in Euclidean space.
result Ruled and translation surfaces are cylindrical.
In this study we give definitions and characterizations of transversal surfaces of timelike ruled surfaces. We study some special cases such as the striction curve is a geodesic, an asymptotic line or a line of curvature. Moreover, we obtain developable conditions for transversal surfaces of a timelike ruled surface.
The study characterizes constant curvature manifolds using ruled surfaces.
problem Characterizing manifolds of constant curvature using ruled surfaces.
method Investigating ruled surfaces in 3d Riemannian manifolds, finding stiction curve, distribution parameter, and fundamental forms.
result Identifies necessary and sufficient conditions for extrinsically flat surfaces to be ruled and proves manifold properties.
Motivated by a number of recent investigations, we define and investigate the various properties of the ruled surfaces depend on three dimensional Lie groups with a bi-variant metric. We give useful results involving the characterizations of these ruled surfaces. Some special ruled surfaces such as normal surface, bino…
In this paper, using the classifications of timelike and spacelike ruled surfaces, we define and study the Mannheim offsets of spacelike ruled surfaces in Minkowski 3-space. We give the conditions for spacelike offset surfaces to be developable.
We study ruled surfaces in R3 which are obtained from dual spher- ical indicatrix curves of dual Frenet vector fields. We find the Gaussian and mean curvatures of the ruled surfaces and give some results of being Wein- garten surface.
Study ruled surfaces in 3D Riemannian manifolds, determining curvature and striction curves.
problem Characterize ruled surfaces in 3D Riemannian manifolds.
method Determine extrinsic and sectional curvature, introduce Sannia frame, characterize striction curves.
result Prove existence and uniqueness of striction curves in space forms, disprove in others.