We prove that every quaternionic-contact structure can be embedded in a quaternionic manifold and define a second fundamental form for a such embedding.
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Quaternion embeddings model entities and relations in knowledge graphs.
problem Modeling latent inter-dependencies and expressive rotations in knowledge graphs.
method Quaternion-valued embeddings and rotations in hypercomplex space.
result Quaternion embeddings achieve state-of-the-art performance on knowledge graph benchmarks.
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 -quaternionic manifolds. Moreover, …
We prove that the embedding of the quaternionic hyperbolic disc into quaternionic hyperbolic -space is tight and thereby obtain the value of the Gromov norm of the quaternionic Kähler class.
We derive an explicit lower bound on the radius of a ball embedded in a quaternionic hyperbolic manifold.
Unified framework for complex-valued eigenfunctions on Riemannian symmetric spaces.
problem Finding a unified scheme for complex-valued eigenfunctions on Riemannian symmetric spaces.
method Employing the Cartan embedding for classical compact Riemannian symmetric spaces and quaternionic Grassmannians.
result Construction of new eigenfunctions on quaternionic Grassmannians.
We study the Penrose transform for the `quaternionic objects' whose twistor spaces are complex manifolds endowed with locally complete families of embedded Riemann spheres with positive normal bundles.
We describe explicitly all quaternionic contact hypersurfaces (qc-hypersurfaces) in the flat quaternion space $\Hnn$ and the quaternion projective space. We show that up to a quaternionic affine transformation a qc-hypersurface in $\Hnn$ is contained in one of the three qc-hyperquadrics in $\Hnn$. Moreover, we show tha…
The study classifies hypersurfaces in quaternionic space forms with constant principal curvatures.
problem Classifying hypersurfaces in quaternionic space forms with specific curvature properties.
method Analyzing curvature-adapted real hypersurfaces in non-flat quaternionic space forms and .
result Classification of hypersurfaces including geodesic hyperspheres, tubes, and specific examples in and .
A complete solution to the quaternionic contact Yamabe problem on the seven dimensional sphere is given. Extremals for the Sobolev inequality on the seven dimensional Hesenberg group are explicitly described and the best constant in the Folland-Stein embedding theorem is determined.
QGNN uses Quaternion space for better graph and node classification.
problem Existing GNN methods struggle with Euclidean vector space limitations.
method Proposes QGNN to learn graph representations in Quaternion space.
result Obtains state-of-the-art results on graph and node classification benchmarks.
We characterise, in the setting of the Kodaira-Spencer deformation theory, the twistor spaces of (co-)CR quaternionic manifolds. As an application, we prove that, locally, the leaf space of any nowhere zero quaternionic vector field on a quaternionic manifold is endowed with a natural co-CR quaternionic structure. Also…
We describe the (complex) quaternionic geometry encoded by the embeddings of the Riemann sphere, with nonnegative normal bundles.
By use of H. C. Wang's bound on the radius of a ball embedded in the fundamental domain of a lattice of a semisimple Lie group, we construct an explicit lower bound for the volume of a quaternionic hyperbolic orbifold that depends only on dimension.
In this paper, we obtain analogues of Jorgensen's inequality for non-elementary groups of isometries of quaternionic hyperbolic -space generated by two elements, one of which is loxodromic. Our result gives some improvement over earlier results of Kim [10] and Markham [15]}. These results also apply to complex hyper…
The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such as the Riemann-Roch Theorem for quaternionic holomorphic vector bundles, the Koda…
We construct new examples of embedded, complete minimal hypersurfaces in quaternionc hyperbolic space and also some minimal foliations. We introduce fans an construct analytic deformations of bisectors.
Paper tackles regularization and sparsification for quaternion neural networks.
problem Regularizing and sparsifying quaternion neural networks for compactness and real-time applications.
method Developed targeted regularization strategies for quaternion neural networks, extending l1 and structured regularization.
result Tailored strategies significantly reduce the number of connections and neurons, resulting in smaller, more compact networks.
Paper proposes a new approach to unify and compare knowledge graph embedding methods.
problem Lack of understanding and comparison of existing knowledge graph embedding methods.
method Introduces a multi-embedding interaction mechanism to unify and generalize existing models.
result Proposes a new multi-embedding model based on quaternion algebra.
We show that every spherical 2-Dupin submanifold that is not a hypersurface is conformally congruent to the standard embedding of the real, complex, quaternionic or octonionic projective plane. We also classify 2-CPC, 2-umbilical and weakly 2-umbilical submanifolds in space forms.
Harmonic morphisms and p-harmonic functions constructed on symmetric spaces.
problem Constructing harmonic morphisms and p-harmonic functions on symmetric spaces.
method Using Cartan embedding and related maps to relate tension field and conformality operator.
result Simple formulae relating tension field and conformality operator on symmetric spaces to those on their images.
We investigate modular embeddings for semi-arithmetic Fuchsian groups. First we prove some purely algebro-geometric or even topological criteria for a regular map from a smooth complex curve to a quaternionic Shimura variety to be covered by a modular embedding. Then we set up an adelic formalism for modular embeddings…
Sharp Veronese rigidity theorem for submanifolds of unit ball.
problem Veronese rigidity of submanifolds under harmonic structure.
method Intrinsic harmonic structure assumptions, Bochner-Gauss mechanism, shape operators.
result Sharp lower bound on maximal normal curvature for specific submanifolds.
Study on quaternionic bisectional curvature for quaternion-Kähler manifolds.
problem Characterize quaternionic bisectional curvature on quaternion-Kähler manifolds.
method Analyzing properties of quaternionic bisectional curvature on specific manifolds.
result Non-negative quaternionic bisectional curvature is only on quaternionic projective space.
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.
problem Conditions for quaternionic structures on symmetric spaces.
method Analysis of Lie group actions and representations.
result Symmetric spaces have invariant quaternionic structures under specific conditions.
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.
problem No specific problem stated; focuses on definition and properties.
method Introduced a modified Dirac operator to define quaternionic k-vector fields.
result Calculated the dimension of quaternionic k-vector fields on .
The paper proves quaternion projective space is unstable.
problem Stability of quaternion projective space.
method Analyzing index of identity map on quaternion space forms.
result Quaternion projective space is unstable.
The paper studies quaternionic structures on GKM graphs and their relation to torus actions on quaternionic projective spaces.
problem Understanding quaternionic structures on GKM graphs and their implications for torus actions.
method Introducing quaternionic structures on GKM graphs and analyzing their properties in the context of torus actions.
result Abstract GKM graphs with specific 2-face structures correspond to torus actions on quaternionic projective spaces or Grassmannians.
Penrose's two-spinor notation for -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 , of dimension , of a quaternionic manifold…
The paper extends Gray's result to quaternion-Kähler manifolds.
problem Understanding quaternion-Kähler manifolds with non-negative quaternionic sectional curvature.
method Introducing quaternionic sectional curvature, proving Wolf spaces have non-negative curvature, and using nearly Kähler twistor spaces.
result Every quaternion-Kähler manifold with non-negative quaternionic sectional curvature is a Wolf space.
Embeds skein algebras into quantum tori using Dehn-Thurston coordinates.
problem Studying representations of Kauffman bracket skein algebras at roots of unity.
method Using the action of the skein algebra on the skein module of the handlebody.
result Explicit reconstruction of unique representation with fixed classical shadow.
Quaternionic curves with specific torsion properties don't exist.
problem Existence of quaternionic Bertrand curves with non-zero torsion and bitorsion.
method Definition of quaternionic (1,3)-Bertrand curves using Type 2-Quaternionic Frame and Matsuda-Yorozu method.
result No quaternionic Bertrand curves with non-zero torsion and bitorsion 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…
Quaternionic differential geometry expands geometric concepts using quaternions.
problem Generalizing geometric concepts to quaternionic constraints.
method Generalizing curves and surfaces, curvature, torsion, differential forms, and directional derivatives to quaternionic constraints.
result Quaternionic formalism provides a suitable language for differential geometry.
Quaternionic CR Geometrymath.DG
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…
The paper classifies compact affine quaternionic curves and surfaces.
problem Classifying compact affine quaternionic curves and surfaces.
method Affine quaternionic manifolds, Kodaira Theorem, fundamental groups, Lie Groups.
result Only quaternionic tori and primary Hopf surface S^3 x S^1 are compact affine quaternionic curves.
The paper explores quaternionic curves using differential geometry.
problem Understanding quaternionic curves.
method Differential geometry applied to quaternionic curves.
result Simpler formulations of quaternionic curves.
Quaternionic Brownian motion on flag manifold linked to sphere diffusion.
problem Modeling quaternionic stochastic areas on quaternionic flag manifolds.
method Relating quaternionic Brownian motion to symplectic Brownian motion and using radial dynamics.
result Quaternionic stochastic areas follow a multivariate normal distribution.
Study cohomology of quaternionic foliations and orbifolds.
problem Understanding cohomology of quaternionic foliations and orbifolds.
method Definition and proof of foliated versions of classical results for quaternionic Kähler manifolds.
result Formulation and proof of foliated versions of classical results for quaternionic Kähler manifolds.
Study counts and equidistributes rational points in quaternionic Heisenberg groups.
problem Counting and equidistribution of rational points in quaternionic Heisenberg groups.
method Arithmetic group actions on quaternionic hyperbolic spaces, Mertens counting formula, Neville equidistribution theorem.
result Proved Mertens counting formula and Neville equidistribution theorem for rational points over definite quaternion algebras.
We apply the general theory of codimension one integrability conditions for -structures developed in arXiv:1306.6817v3 [math.DG] to the case of quaternionic CR geometry. We obtain necessary and sufficient conditions for an almost CR quaternionic manifold to admit local immersions as an hypersurface of the quaternion…
A partial solution of the quaternionic contact Yamabe problem on the quaternionic sphere is given. It is shown that the torsion of the Biquard connection vanishes exactly when the trace-free part of the horizontal Ricci tensor of the Biquard connection is zero and this occurs precisely on 3-Sasakian manifolods. All con…
A tensor invariant is defined on a quaternionic contact manifold in terms of the curvature and torsion of the Biquard connection involving derivatives up to third order of the contact form. This tensor, called quaternionic contact conformal curvature, is similar to the Weyl conformal curvature in Riemannian geometry an…