arXiv research

Classifies real rational knots and curves in a specific quadric space.

Study holomorphic isometric embeddings of a Grassmannian into quadrics.

We study CR quadrics satisfying a symmetry property $(\tilde S)$ which is slightly weaker than the symmetry property $(S)$, 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…

Generalizes embedding complex Grassmannians into quadrics.

Study finds Stäckel equivalence for superintegrable systems via invariant quadrics.

Study isotropic curves on complex quadric with geometric relations.

Transformed quadrics from 2D to higher dimensions.

In this paper, we study ruled surfaces and quadrics in the 3-dimensional Euclidean space which are of finite $III$-type, that is, they are of finite type, in the sense of B.-Y. Chen, with respect to the third fundamental form. We show that helicoids and spheres are the only ruled and quadric surfaces of finite $III$-ty…

This study classifies quadric surfaces in 3-sphere as Weingarten surfaces.

The paper classifies and determines properties of specific hypersurfaces in complex hyperbolic quadrics.

Study of discrete Koenigs nets and their properties.

The study characterizes quadrics among affine hyperspheres based on centroid collinearity of sections.

We develop an essentially algebraic method to study biharmonic curves into an implicit surface. Although our method is rather general, it is especially suitable to study curves into surfaces defined by a polynomial equation: in particular, we use it to give a complete classification of biharmonic curves into real quadr…

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 …

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.

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 …

Minimal Lagrangian surfaces in complex hyperbolic quadric via loop group method.

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.

Researchers find explicit Bäcklund transforms for specific quadrics.

Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics which are isometric along asymptotic lines and …

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.

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…

Study on GL(2) geometries on complex manifolds, focusing on Kähler-Einstein and Fano manifolds.

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$^{\mathrm{th}}$ 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…

Paper proves non-existence of certain hypersurfaces in complex quadric.

Here are described the geometric structures of the lines of principal curvature and the partially umbilic singularities of the tridimensional non compact generic quadric hypersurfaces of ${\mathbb R}^4$. This includes the ellipsoidal hyperboloids of one and two sheets and the toroidal hyperboloids. The present study co…

The first part of this article is devoted to the study families of totally real intersecting $n$-submanifolds of $(\Bbb C^n,0)$. We give some conditions which allow to straighten holomorphically the family. If this is not possible to do it formally, we construct a germ of complex analytic set at the origin which intere…

Describes geodesic scattering on hyperboloids using quadrics results.

Study on real hypersurfaces in complex quadric with special connections and operators.

We illustrate the theory of one-dimensional pluri-Lagrangian systems with the example of commuting billiard maps in confocal quadrics.

We investigate the relation between quadrics and their Christoffel duals on the one hand, and certain zero mean curvature surfaces and their Gauss maps on the other hand. To study the relation between timelike minimal surfaces and the Christoffel duals of 1-sheeted hyperboloids we introduce para-holomorphic elliptic fu…

Study fibrations of projective spaces for maximal representations.

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 $\mathbb{C}^3$ of 2-dimensional general quadrics with common conjugate system given by the spherical coordinates on the complex sphere $\…

Study shows K-moduli spaces of curves on quadrics and K3 surfaces match with VGIT quotients.

Defines CAMC discrete nets and their properties.

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.

This paper proves integrability of Birkhoff billiards inside convex cones.

The quadric ansatz solves dKP equations in arbitrary dimensions, leading to Einstein-Weyl structures.

Study Lagrangian submanifolds on complex hyperbolic quadric.

Adapts stereographic projection for ellipsoid and elliptic paraboloid.

We investigate basic features of Bianchi's Bäcklund transformation of quadrics to see if it can be obtained under weaker assumptions and if it can be generalized to deformations of other 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…