arXiv research

Quaternionic differential geometry expands geometric concepts using quaternions.

The paper explores quaternionic curves using differential geometry.

Study describes moduli of quaternionic hyperbolic triples of points.

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 develops quaternionic toric geometry and classifies local actions.

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

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…

Motivated by the analogies between the projective and the almost quaternionic geometries, we study the generalized planar curves and mappings. We follow, recover, and extend the classical approach as developed by Mikes and Sinyukov. Then we exploit the impact of the general results in the almost quaternionic geometry. …

The paper proves new curvature estimates in quaternionic contact geometry.

This is a survey on quaternion Hermitian Weyl (locally conformally quaternion Kähler) and hyperhermitian Weyl (locally conformally hyperkähler) manifolds. These geometries appear by requesting the compatibility of some quaternion Hermitian or hyperhermitian structure with a Weyl structure. The motivation for such a stu…

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

The fibre bundles adjoint to generalized almost quaternionic structures are studied. The most important classes of generalized almost quaternionic manifolds are considered.

Study of quaternionic hyperbolic space bisectors and their decompositions.

We study hyperkahler cones and their corresponding quaternion-Kahler spaces. We present a classification of 4(n-1)-dimensional quaternion-Kahler spaces with n abelian quaternionic isometries, based on dualizing superconformal tensor multiplets. These manifolds characterize the geometry of the hypermultiplet sector of p…

Solves a specific Calabi conjecture on special nilmanifolds.

The paper studies metrics and geodesics on a quaternionic Heisenberg group.

Differential calculus on the quantum quaternionic group GL(1,H$_q$) is introduced.

We give an overview of some recent results in hypersymplectic and para-quaternionic Kahler geometry, and introduce the notion of split three-Sasakian manifold. In particular, we discuss the twistor spaces and Swann bundles of para-quaternionic Kahler manifolds. These are used to classify examples with a fully homogeneo…

The generalized Feix--Kaledin construction shows that c-projective $2n$-manifolds with curvature of type $(1,1)$ are precisely the submanifolds of quaternionic $4n$-manifolds which are fixed points set of a special type of quaternionic $S^1$ action $v$. In this paper, we consider this construction in the presence of in…

New proof for quaternionic structures on specific manifolds via automorphisms.

We apply the general theory of codimension one integrability conditions for $G$-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…

Following the Cartans's original method of equivalence supported by methods of parabolic geometry, we provide a complete solution for the equivalence problem of quaternionic contact structures, that is, the problem of finding a complete system of differential invariants for two quaternionic contact manifolds to be loca…

Study cohomology of quaternionic foliations and orbifolds.

Study of real and quaternionic Lie algebroid connections on manifolds.

We exploit the Cartan-Kähler theory to prove the local existence of real analytic quaternionic contact structures for any prescribed values of the respective curvature functions and their covariant derivatives at a given point on a manifold. We show that, in a certain sense, the different real analytic quaternionic con…

We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension $2$. We prove a Mertens counting formula for the rational points over a definite quat…

New universal invariant operators are introduced in a class of geometries which include the quaternionic structures and their generalisations as well as 4-dimensional conformal (spin) geometries. It is shown that, in a broad sense, all invariants and invariant operators arise from these universal operators and that the…

Any oriented 4-dimensional real vector bundle is naturally a line bundle over a bundle of quaternion algebras. In this paper we give an account of modules over bundles of quaternion algebras, discussing Morita equivalence, characteristic classes and K-theory. The results have been used to describe obstructions for the …

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, …

The paper establishes sub-gradient estimates and entropy formulas for quaternionic contact geometry heat equations.

New submersions found in nearly Kähler geometry.

We introduce the notion of even Clifford structures on Riemannian manifolds, a framework generalizing almost Hermitian and quaternion-Hermitian geometries. We give the complete classification of manifolds carrying parallel even Clifford structures: Kähler, quaternion-Kähler and Riemannian products of quaternion-Kähler …

New sigma models compute graviton scattering amplitudes from quaternionic geometry.

We clarify the explicit structure of the Hurwitz quaternion order, which is of fundamental importance in Riemann surface theory and systolic geometry.

An important problem in quaternionic hyperbolic geometry is to classify ordered $m$-tuples of pairwise distinct points in the closure of quaternionic hyperbolic n-space, $\overline{{\bf H}_\bh^n}$, up to congruence in the holomorphic isometry group ${\rm PSp}(n,1)$ of ${\bf H}_\bh^n$. In this paper we concentrate on tw…

This paper studies geometric structures on manifolds with specific symplectic properties.

The paper classifies hypercomplex Lie algebras and solvmanifolds using quaternionic Jordan form.

Investigates special metrics in hypercomplex geometry.

We study the moduli space of quaternionic Kaehler structures on a compact manifold of dimension 4n (n>2) from a point of view of Riemannian geometry, not twistor theory. Then we obtain a rigidity theorem for quaternionic Kaehler structures of nonzero scalar curvature by observing the moduli space.

Study on null submanifolds in indefinite complex contact geometry.

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.

The paper establishes correspondences between quaternionic spinors, Minkowski flags, and hyperbolic horospheres.

The moduli space of the Calabi-Yau three-folds, which play a role as superstring ground states, exhibits the same {\em special geometry} that is known from nonlinear sigma models in $N=2$ supergravity theories. We discuss the symmetry structure of special real, complex and quaternionic spaces. Maps between these spaces…

Notions of self-dual and anti self-dual almost quaternionic structures are introduced. The complete classification of self-dual and anti self-dual generalized Kaehler manifolds is obtained.

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 introduce the notion of tame $ρ$-quaternionic manifold that permits the construction of a finite family of $ρ$-connections, significant for the geometry involved. This provides, for example, the following: (1) a new simple global characterisation of flat (complex-)quaternionic manifolds, and (2) a new simple constru…

We explicitly describe all SO(7)-invariant almost quaternion-Hermitian structures on the twistor space of the six sphere and determine the types of their intrinsic torsion.

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…