Study on null curves in Klein's quartic moduli space.
problem Existence of minimal surfaces with specific planar ends.
method Methods developed by Robert Bryant.
result Minimal surfaces with 9 embedded planar ends do not exist.
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48 results for “Klein quadric”
New method constructs AdS 3-manifolds and applies to Higgs bundles and minimal immersions.
problem Understanding AdS 3-manifolds and their connections to Higgs bundles and minimal immersions.
method Developed a new construction method for AdS structures.
result Recovered Tholozan's formula for AdS 3-manifold volumes and characterized representations for minimal immersions.
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.
The theory of slice regular functions of a quaternion variable is applied to the study of orthogonal complex structures on domains Ω of R^4. When Ω is a symmetric slice domain, the twistor transform of such a function is a holomorphic curve in the Klein quadric. The case in which Ω is the complement of a parabola is st…
Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spacesmath.DG
The paper develops RM frames for isotropic and pseudo-isotropic spaces and characterizes spherical curves.
problem Differential geometry of curves in isotropic and pseudo-isotropic 3-spaces.
method Developing rotation minimizing frames and applying them to spherical curves in isotropic and pseudo-isotropic spaces.
result Characterization of spherical curves via linear equations involving curvatures and RM frames.
Transformed quadrics from 2D to higher dimensions.
problem Generalizing quadric transformations to higher dimensions.
method Bianchi's Hazzidakis transformation method.
result Generalization to higher dimensional quadrics.
We study CR quadrics satisfying a symmetry property which is slightly weaker than the symmetry property , recently introduced by W. Kaup, which requires the existence of an automorphism reversing the gradation of the Lie algebra of infinitesimal automorphisms of the quadric. We characterize quadrics s…
Billiards in confocal quadrics show pluri-Lagrangian systems in action.
problem Understanding billiard dynamics in specific geometric shapes.
method Illustration of one-dimensional pluri-Lagrangian systems with confocal quadrics.
result Commuting billiard maps demonstrate pluri-Lagrangian systems.
The study constructs non-planar nets and webs on quadrics and Minkowski space.
problem Constructing non-planar nets and webs on quadrics and Minkowski space.
method Introducing canonical parametrisations, exploiting connections with classical deformations, and using Laguerre geometric notions.
result Existence and construction of octahedral grids and webs of surfaces.
Classifies real rational knots and curves in a specific quadric space.
problem Classifying real rational knots and curves in a quadric space of signature .
method Classification through a study of real rational curves of low degree in the quadric.
result Provides representatives of all real rational knots of degree in the quadric.
In the present article, we provide examples of fake quadrics, that is, minimal complex surfaces of general type with the same numerical invariants as the smooth quadric in $\PP ^3$ which are quotients of the bidisc by an irreducible lattice of automorphisms. Moreover, we list classes of arithmetic lattices over a real …
Study finds Stäckel equivalence for superintegrable systems via invariant quadrics.
problem Understanding Stäckel equivalence in superintegrable systems.
method Using invariant quadrics to determine Stäckel classes of superintegrable systems.
result Stäckel classes of superintegrable systems can be derived from associated invariant quadrics.
A fake quadric is a smooth projective surface that has the same rational cohomology as a smooth quadric surface but is not biholomorphic to one. We provide an explicit classification of all irreducible fake quadrics according to the commensurability class of their fundamental group. To accomplish this task, we develop …
Extends Moutard quadric concept to higher dimensions.
problem Higher order contact of quadrics with surfaces in 3D.
method Extension to hypersurfaces in arbitrary dimensions.
result Extension of Moutard quadric concept to higher dimensions.
We provide a generalization of Bianchi's Bäcklund transformation from 2-dimensional quadrics to higher dimensional quadrics. The starting point of our investigation is the higher dimensional (infinitesimal) version of Bianchi's main four theorems on the theory of deformations of quadrics and Bianchi's treatment of the …
The study identifies helicoids and spheres as the only finite III-type ruled and quadric surfaces.
problem Characterizing finite III-type ruled and quadric surfaces.
method Analyzing surfaces in 3D Euclidean space with respect to the third fundamental form.
result Helicoids and spheres are the only finite III-type ruled and quadric surfaces.
Generalizes embedding complex Grassmannians into quadrics.
problem Holomorphic isometric embeddings of complex Grassmannians into quadrics.
method Generalization of do Carmo-Wallach theory for moduli spaces.
result Moduli spaces of embeddings discussed.
Study holomorphic isometric embeddings of a Grassmannian into quadrics.
problem Holomorphic isometric embeddings of complex Grassmannian into quadrics.
method Generalization of do Carmo--Wallach theory to study moduli space.
result Moduli space of embeddings up to equivalence discussed.
In trying to provide explicit deformations of quadrics the starting point of our investigation is to use Bianchi's link between real deformations of totally real regions of real paraboloids and various totally real forms of the sine-Gordon equation coupled with Bianchi's simple observation that the vacuum soliton of th…
The paper classifies Hopf hypersurfaces in complex quadrics with commuting Jacobi operators.
problem Characterizing Hopf real hypersurfaces with commuting Jacobi operators.
method Investigating the commuting property between normal and structure Jacobi operators.
result A remarkable classification of Hopf real hypersurfaces in the complex quadric with commuting Jacobi operators.
Study classifies hypersurfaces with parallel structure Jacobi operator in complex quadric.
problem Characterizing real hypersurfaces with specific geometric properties.
method Defined and classified real hypersurfaces in complex quadric with parallel structure Jacobi operator.
result Complete classification of real hypersurfaces with parallel structure Jacobi operator.
The paper classifies real hypersurfaces with a special structure in complex quadrics.
problem Characterizing real hypersurfaces with specific geometric properties in complex quadrics.
method Derived formulas for structure Jacobi operator and its derivative, classified hypersurfaces with Reeb parallel structure.
result Complete classification of Hopf real hypersurfaces with Reeb parallel structure Jacobi operator.
Researchers find explicit Bäcklund transforms for specific quadrics.
problem Isometric deformations of diagonal higher dimensional quadrics without center.
method Explicitly found Bäcklund transforms using the Bianchi Permutability Theorem and 3-moving Möbius configuration.
result Explicit solutions can be iterated with arbitrary constants.
Real hypersurfaces in the complex hyperbolic quadric with Reeb parallel structure Jacobi operatormath.DG
Study classifies real hypersurfaces with special Jacobi operator in complex hyperbolic quadric.
problem Characterizing real hypersurfaces with specific properties in complex hyperbolic quadric.
method Introduced Reeb parallel structure Jacobi operator and classified real hypersurfaces.
result Classification of real hypersurfaces with Reeb parallel structure Jacobi operator.
It was observed by Tod and later by Dunajski and Tod that the Boyer-Finley (BF) and the dispersionless Kadomtsev-Petviashvili (dKP) equations possess solutions whose level surfaces are central quadrics in the space of independent variables (the so-called central quadric ansatz). It was demonstrated that generic solutio…
The paper classifies Hopf hypersurfaces with constant curvatures on complex quadrics.
problem Classifying Hopf hypersurfaces with specific curvature properties.
method Analyzing hypersurfaces on complex quadrics with at most five distinct constant principal curvatures.
result All classified hypersurfaces are open parts of homogeneous examples.
Classifies real algebraic curves on a quadric ellipsoid of specific degree.
problem Classifying real algebraic curves of bidegree (5,5) on the quadric ellipsoid.
method Reduction to curves in the second Hirzebruch surface, combining classical construction methods on toric surfaces.
result Previously known restrictions form a complete system for this bidegree.
We establish a link between Archimedes' method of integration for calculating areas, volumes and centers of mass of segments of parabolas and quadrics of revolution by factorization via the moments of a balance and an integration technique for a particular integrable system, namely Bianchi's Bäcklund transformation for…
We discuss holomorphic isometric embeddings of the projective line into quadrics using a generalisation of the theorem of do Carmo--Wallach to provide a description of their moduli spaces up to image and gauge--equivalence. Moreover, we show rigidity of the real standard map from the projective line into quadrics.
Confocal quadrics capture (encode) and geometrize spectral properties of symmetric operators. Certain metric-projective properties of confocal quadrics (most of them established in the first half of the XIX century) {\it carry out} (stick and transfer) by rolling to and influence surfaces {\it applicabl…
We prove here that when all planes transverse and nearly perpendicular to the axis of a surface of revolution intersect it in loops having central symmetry, the surface must be quadric. It follows that the quadrics are the only surfaces of revolution without skewloops. Similar statements hold for hypersurfaces of revol…
Study classifies real hypersurfaces in complex quadric with special tensor property.
problem Characterizing real hypersurfaces in complex quadric with commuting Ricci tensor.
method Introduced and utilized the concept of commuting Ricci tensor to classify hypersurfaces.
result Complete classification of real hypersurfaces in with commuting Ricci tensor.
The study classifies real hypersurfaces in complex hyperbolic quadrics with isometric Reeb flow.
problem Classifying real hypersurfaces with isometric Reeb flow in complex hyperbolic quadrics.
method Classification based on the properties of the hypersurfaces and their embeddings.
result The existence and properties of real hypersurfaces with isometric Reeb flow are classified, leading to the non-existence in odd-dimensional cases.
Paper proves non-existence of certain hypersurfaces in complex quadric.
problem Non-existence of Hopf real hypersurfaces with parallel normal Jacobi operator.
method Introducing -parallel and Reeb parallel normal Jacobi operators, proving non-existence theorems.
result Non-existence of Hopf real hypersurfaces with -parallel normal Jacobi operator.
Study isotropic curves on complex quadric with geometric relations.
problem Characterize isotropic curves on the complex quadric.
method Analyze geometric properties and relations of isotropic curves.
result Discovers relations between isotropic curves and surfaces in spaceforms.
Describes geodesic scattering on hyperboloids using quadrics results.
problem Understanding geodesic scattering on hyperboloids.
method Uses results from Moser and Knörrer on quadrics and Neumann system.
result Extends Knörrer's map to the projective closure of hyperboloids.
The paper classifies and determines properties of specific hypersurfaces in complex hyperbolic quadrics.
problem Analyzing Hopf hypersurfaces with constant principal curvatures in complex hyperbolic quadrics.
method Classification and determination of principal curvatures for hypersurfaces with different numbers of distinct curvatures.
result Classification and determination of principal curvatures for Hopf hypersurfaces with up to four distinct values.
Quadrics and Scherk towersmath.DG
Investigates the connection between quadrics and Christoffel duals, and zero mean curvature surfaces.
problem Understanding the relationship between quadrics and Christoffel duals, and zero mean curvature surfaces.
method Introducing para-holomorphic elliptic functions to study timelike minimal surfaces and their Christoffel duals of 1-sheeted hyperboloids.
result Curves of type change for real isothermic surfaces of mixed causal type are aligned with the real curvature line net.
This study classifies quadric surfaces in 3-sphere as Weingarten surfaces.
problem Extension of quadric surfaces of revolution to 3-sphere.
method Rigorous classification and characterization using spherical angular momentum.
result Spherical ellipsoids, hyperboloids, and paraboloids are Weingarten surfaces with a specific cubic relation between principal curvatures.
On a discretization of confocal quadrics. II. A geometric approach to general parametrizationsmath.DG
A geometric approach discretizes confocal quadrics, leading to novel discrete nets.
problem Discretization of confocal quadrics.
method Geometric characterization and factorizable orthogonal coordinate systems.
result Explicit computation of coordinate functions for discrete confocal quadrics.
New proof classifies contact real hypersurfaces in complex hyperbolic quadric.
problem Classifying contact real hypersurfaces with constant mean curvature.
method New proof using tubes and horospheres.
result Contact real hypersurfaces are congruent to tubes or horospheres.
Geodesic and billiard flows on quadrics in pseudo-Euclidean spaces: L-A pairs and Chasles theoremnlin.SI
In this article we construct L--A representations of geodesic flows on quadrics and of billiard problems within ellipsoids in the pseudo--Euclidean spaces. A geometric interpretation of the integrability analogous to the classical Chasles theorem for symmetric ellipsoids is given. We also consider a generalization of t…
We provide the first explicit examples of deformations of higher dimensional quadrics: a straightforward generalization of Peterson's explicit 1-dimensional family of deformations in of 2-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere $\…
Defines CAMC discrete nets and their properties.
problem Understanding CAMC discrete nets and their properties.
method Defining CAMC discrete nets and proving properties.
result Properties of CAMC discrete nets are equivalent to properties of compatible interpolating quadrics.
The paper characterizes projective spaces and quadrics using strictly nef bundles.
problem Characterizing projective spaces and quadrics using geometric properties of bundles.
method Analyzing strictly nef bundles on smooth projective varieties and curves.
result Strictly nef bundles on smooth projective varieties and curves have specific geometric properties.
New tensor introduced for complex quadric hypersurfaces, no Hopf hypersurfaces found.
problem Characterizing real hypersurfaces in complex quadric using star-Ricci tensors.
method Introducing and analyzing star-Ricci tensors in real hypersurfaces of complex quadric.
result No Hopf hypersurfaces exist in , with certain star-Ricci tensors.
We provide a generalization of Bianchi's triply conjugate systems containing a family of deformations of 2-dimensional quadrics together with its Bäcklund transformation to higher dimensions.
New exotic 4-manifolds created from lines and quadrics in CP^2.
problem Creating new exotic 4-manifolds homeomorphic but not diffeomorphic to CP^2 # 8 \overline{CP^2} and CP^2 # 9 \overline{CP^2}.
method Rational blowdown surgery along 4-valent plumbing graphs formed by complex lines and quadrics in CP^2.
result Graph classes from \cite{weighted} have representatives admitting rational blowdown leading to exotic manifolds.