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.
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48 results for “projective quadric”
Adapts stereographic projection for ellipsoid and elliptic paraboloid.
problem Projecting quadric surfaces using stereographic method.
method Adapted stereographic projection for ellipsoid and elliptic paraboloid, analyzing geometric properties and challenges.
result Established results on eccentricities, curvatures, arc length, and areas of intersections and projections.
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.
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…
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 …
The study identifies only two rational surfaces with constant scalar curvature Kähler metrics.
problem Finding rational surfaces with constant scalar curvature Kähler metrics.
method Analyzing projective rational surfaces and their Kähler metrics.
result There are only two rational surfaces (projective plane and quadric surface) with constant scalar curvature Kähler metrics.
Study fibrations of projective spaces for maximal representations.
problem Characterize maximal representations of surface groups.
method Analyze fibrations of projective spaces and use geometric structures.
result Maximal representations correspond to fibrations with specific properties.
In this paper, we show that if the tangent bundle of a smooth projective variety is strictly nef, then it is isomorphic to a projective space; if a projective variety has strictly nef , then it is isomorphic to or quadric . We also prove that on elliptic curves, strict…
Our main aim is to provide a uniform geometric characterization of the analogues over arbitrary fields of the four complex Severi varieties, i.e.~the quadric Veronese varieties in 5-dimensional projective spaces, the Segre varieties in 8-di\-men\-sional projective spaces, the line Grassmannians in 14-dimensional projec…
Discretizes projective minimal surfaces using geometric characterizations.
problem Classifying discrete projective minimal surfaces.
method Introduced canonical discrete models and line congruences.
result Discrete analogues of classical Lie quadrics and surfaces.
We classify real hypersurfaces with isometric Reeb flow in the complex quadrics Q^m for m > 2. We show that m is even, say m = 2k, and any such hypersurface is an open part of a tube around a k-dimensional complex projective space CP^k which is embedded canonically in Q^{2k} as a totally geodesic complex submanifold. A…
New complex orthogonal structures found on a 3D manifold.
problem Finding new geometric structures on a 3D manifold.
method Constructing uniformizable complex orthogonal and projective structures.
result A manifold has multiple uniformizable complex orthogonal structures.
We present the theory of tensors with Young tableau symmetry as an efficient computational tool in dealing with the polynomial first integrals of a natural system in classical mechanics. We relate a special kind of such first integrals, already studied by Lundmark, to Beltrami's theorem about projectively flat Riemanni…
The article interprets Darboux's surfaces using triality in differential geometry.
problem Understanding Darboux's surfaces through triality.
method Geometric interpretation of Darboux's surfaces using triality.
result Darboux's surfaces are related to totally isotropic surfaces in a 6D projective quadric.
The present article studies the class of Einstein-Hermitian harmonic maps of constant Kaehler angle from the projective line into quadrics. We provide a description of their moduli spaces up to image, and gauge-equivalence using the language of vector bundles and representation theory. It is shown that the dimension of…
We describe some general constructions on a real smooth projective 4-quadric which provide analogues of the Willmore functional and conformal Gauss map in both Lie sphere and projective differential geometry. Extrema of these functionals are characterized by harmonicity of this Gauss map.
This is an expanded and updated version of a lecture series I gave at Seoul National University in September 1997. It is in some sense an update of the 1979 Griffiths and Harris paper with a similar title. I discuss: Homogeneous varieties, Topology and consequences Projective differential invariants, Varieties with deg…
AQFC method estimates mesh curvatures using quadratic surfaces.
problem Estimating curvatures for irregular polygonal meshes.
method Local approximation of vertices and normals by quadratic surfaces, computed as implicit surfaces.
result AQFC provides robust curvature estimation for irregular meshes.
Researchers found a geometric duality for systems of conservation laws.
problem Systems of conservation laws and their additional conservation laws.
method Assigning ruled surfaces in projective space and defining dual systems.
result Hamiltonian systems are autodual, and 3-component nondiagonalizable systems are dual to systems with constant characteristic speeds.
Construct noncommutative deformations of algebraic submanifolds in R^n.
problem Deforming algebraic submanifolds in noncommutative geometry.
method Using twisted differential geometry and Drinfel'd twists, constructing noncommutative deformations of algebraic submanifolds.
result Explicitly worked out deformations of quadrics in R^3.
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.
The paper studies smooth projective varieties with strictly nef tangent bundles and their properties.
problem Understanding smooth projective varieties with strictly nef tangent bundles.
method Analyzing the tangent bundle properties and using rational connectedness.
result Smooth projective varieties with strictly nef tangent bundles are rationally connected.
Study curvature loci of 3-manifolds in R^6 and R^5.
problem Characterize curvature loci of 3-manifolds in different dimensions.
method Refine affine classification of real nets of quadrics, study singularities, and analyze systems of ternary cubics.
result Obtain generic curvature loci and singularities of 3-manifolds.
Motivated by Gray's work on tube formulae for complex submanifolds of complex projective space equipped with the Fubini-Study metric, Riemannian foliations of projective space are studied. We prove that there are no complex Riemannian foliations of any open subset of of codimension one. As a consequence …
We construct N=2 supersymmetric nonlinear sigma models whose target spaces are tangent as well as cotangent bundles over the quadric surface Q^{n-2} = SO(n)/[SO(n-2)\times U(1)]. We use the projective superspace framework, which is an off-shell formalism of N=2 supersymmetry.
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…
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 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.
We show that on a surface locally every affine torsion-free connection is projectively equivalent to a Weyl connection. First, this is done using exterior differential system theory. Second, this is done by showing that the solutions of the relevant PDE are in one-to-one correspondence with the sections of the `twistor…
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 …
It is demonstrated that the stationary Veselov-Novikov (VN) and the stationary modified Veselov-Novikov (mVN) equations describe one and the same class of surfaces in projective differential geometry: the so-called isothermally asymptotic surfaces, examples of which include arbitrary quadrics and cubics, quartics of Ku…
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.
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 study shows conditions for Kähler manifolds to have rational cohomology of complex projective space.
problem Conditions for Kähler manifolds to have rational cohomology of complex projective space.
method Analyzing the Calabi curvature operator and its positivity conditions.
result Compact Kähler manifolds with specific curvature conditions have rational cohomology of complex projective space.
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.
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.
Geometry of holomorphic curves from point of view of open Toda systems is discussed. Parametrization of curves related this way to non-exceptional simple Lie algebras is given. This gives rise to explicit formulas for minimal surfaces in real, complex and quaternionic projective spaces or complex quadrics. The paper ge…
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.