arXiv research

Geometric structures on quaternionic unit ball for slice regular Möbius transformations.

Study of Riemannian geometry on quaternionic unit ball linked to Sp(1,1) group.

The paper uses quaternions to model quantum learning on devices.

Smooth manifold structure on Möbius transformations of quaternionic ball identified.

In the present paper we introduce and study a new notion of toric manifold in the quaternionic setting. We develop a construction with which, starting from appropriate $m$-dimensional Delzant polytopes, we obtain manifolds of real dimension $4m$, acted on by $m$ copies of the group ${\rm Sp}(1)$ of unit quaternions. Th…

We study the relations between the quaternion $H$-type group and the boundary of the unit ball on two dimensional quaternionic space. The orthogonal projection of the space of square integrable functions defined on quaternion $H$-type group into its subspace of boundary values of $q$-holomorphic functions is consider. …

QRPNNs use quaternion-valued recurrent correlation neural networks to solve cross-talk issues.

It has long been known that differential forms on complex manifolds can be decomposed under the action of the complex structure to give the Dolbeault complex. This paper presents an analogous double complex for quaternionic manifolds using the fact that the cotangent space is isomorphic to a quaternionic vector space. …

We prove that the multiplication maps $\mathbb{s}^n \times \mathbb{s}^n \rightarrow \mathbb{s}^n$ ($n = 1, 3, 7$) for unit complex, quaternion and octonion numbers are, up to isometries of domain and range, the unique Lipschitz constant minimizers in their homotopy classes. Other geometrically natural maps, such as pro…

In this paper we study the projective automorphism group of domains in real, complex, and quaternionic projective space and present two new characterizations of the unit ball in terms of the size of the automorphism group and the regularity of the boundary.

We study certain polynomial trace identities in the group $SL(2,\IC)$ and their application in the theory of discrete groups. We obtain canonical representations for two generator groups in §4 and then in §5 we give a new proof for Gehring and Martin's polynomial trace identities for good words, and extend that result …

The quaternions are non-commutative. The deviation from commutativity is encapsulated in the commutator of unit quaternions. It is known that the k-th power of the commutator is null-homotopic if and only if k is divisible by 12. The main purpose of this paper is to construct a concrete null-homotopy of the 12-th power…

The purpose of this paper is to give presentations for projective $S$-unit groups of the Hurwitz order in Hamilton's quaternions over the rational field $\mathbb{Q}$. To our knowledge, this provides the first explicit presentations of an $S$-arithmetic lattice in a semisimple Lie group with $S$ large. In particular, we…

The paper explores metrics on Lie groups and their connections to dual quaternions.

Representations of two bridge knot groups in the isometry group of some complete Riemannian 3-manifolds as $E^{3}$ (Euclidean 3-space), $H^{3}$ (hyperbolic 3-space) and $E^{2,1}$ (Minkowski 3-space), using quaternion algebra theory, are studied. We study the different representations of a 2-generator group in which th…

A spacelike surface in the Minkowski 3-space is called a constant slope surface if its position vector makes a constant angle with the normal at each point on the surface. These surfaces completely classified in [J. Math. Anal. Appl. 385 (1) (2012) 208-220]. In this study, we give some relations between split quaternio…

New normalizing flows model molecular crystal structures.

New bounds on diameters and generators for specific lattices and graphs.

Sharp Veronese rigidity theorem for submanifolds of unit ball.

The main goal of this work is to study the sub-Laplacian of the unit sphere which is obtained by lifting with respect to the Hopf fibration the Laplacian of the quaternionic projective space. We obtain in particular explicit formulas for its heat kernel and deduce an expression for the Green function of the conformal s…

Perception is fundamental to many robot application areas especially in service robotics. Our aim is to perceive and model an unprepared kitchen scenario with many objects. We start with the perception of a single target object. The modeling relies especially on fusing and merging of weak information from the sensors o…

Quaternionic approach to conformal superminimal surfaces in four-space

The paper explores CR structures and their leaf spaces in semi-Riemannian manifolds.

Classifies surfaces with great and small circles through each point.

Hypernom is a virtual reality game. The cells of a regular 4D polytope are radially projected to S^3, the sphere in 4D space, then stereographically projected to 3D space where they are viewed in the headset. The orientation of the headset is given by an element of the group SO(3), which is also a space that is double …

Classifications of all biharmonic isoparametric hypersurfaces in the unit sphere, and all biharmonic homogeneous real hypersurfaces in the complex or quaternionic projective spaces are shown. Answers in case of bounded geometry to Chen's conjecture or Caddeo, Montaldo and Piu's one on biharmonic maps into a manifold of…

Study on curvature properties of exotic 7-sphere using quaternionic geometry.

Study on quaternionic bisectional curvature for quaternion-Kähler manifolds.

In this paper, we give the definitions and characterizations of quaternionic Salkowski, quaternionic anti-Salkowski and quaternionic similar curves in the Euclidean spaces E^3 and E^4. We obtain relationships between these curves and some special quaternionic curves such as quaternionic slant helices and quaternionic B…

We construct explicit left invariant quaternionic contact structures on Lie groups with zero and non-zero torsion, and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of quaternionic contact manifolds not locally quaternionic contact conformal to the quaternionic sphere. W…

The study finds conditions for quaternionic structures on symmetric spaces.

We call a quaternionic Kaehler manifold with non-zero scalar curvature, whose quaternionic structure is trivialized by a hypercomplex structure, a hyper-Hermitian quaternionic Kaehler manifold. We prove that every locally symmetric hyper-Hermitian quaternionic Kaehler manifold is locally isometric to the quaternionic p…

We introduce a natural notion of quaternionic map between almost quaternionic manifolds and we prove the following, for maps of rank at least one: 1) A map between quaternionic manifolds endowed with the integrable almost twistorial structures is twistorial if and only if it is quaternionic. 2) A map between quaternion…

Defines quaternionic k-vector fields on quaternionic Kähler manifolds.

The paper proves quaternion projective space is unstable.

The paper studies quaternionic structures on GKM graphs and their relation to torus actions on quaternionic projective spaces.

Penrose's two-spinor notation for $4$-dimensional Lorentzian manifolds can be extended to two-component notation for quaternionic manifolds, which is a very useful tool for calculation. We construct a family of quaternionic complexes over unimodular quaternionic manifolds by elementary calculation. On complex quaternio…

We introduce the notion of CR quaternionic map and we prove that any such real-analytic map, between CR quaternionic manifolds, is the restriction of a quaternionic map between quaternionic manifolds. As an application, we prove, for example, that for any submanifold $M$, of dimension $4k-1$, of a quaternionic manifold…

The paper extends Gray's result to quaternion-Kähler manifolds.

Quaternionic curves with specific torsion properties don't exist.

The conformal infinity of a quaternionic-Kahler metric on a 4n-manifold with boundary is a codimension 3-distribution on the boundary called quaternionic contact. In dimensions 4n-1 greater than 7, a quaternionic contact structure is always the conformal infinity of a quaternionic-Kahler metric. On the contrary, in dim…

Modelled on a real hypersurface in a quaternionic manifold, we introduce a quaternionic analogue of CR structure, called quaternionic CR structure. We define the strong pseudoconvexity of this structure as well as the notion of quaternionic pseudohermitian structure. Following the construction of the Tanaka-Webster con…

Quaternionic differential geometry expands geometric concepts using quaternions.

The paper explores quaternionic curves using differential geometry.

Quaternionic Brownian motion on flag manifold linked to sphere diffusion.

Study cohomology of quaternionic foliations and orbifolds.

Motivated by the quaternionic geometry corresponding to the homogeneous complex manifolds endowed with (holomorphically) embedded spheres, we introduce and initiate the study of the `quaternionic-like manifolds'. These contain, as particular subclasses, the CR quaternionic and the $ρ$-quaternionic manifolds. Moreover, …

Study counts and equidistributes rational points in quaternionic Heisenberg groups.