arXiv research

In this note, we study deformations of quaternionic hyperbolic lattices in larger quaternionic hyperbolic spaces and prove local rigidity results. On the other hand, surface groups are shown to be more flexible in quaternionic hyperbolic plane than in complex hyperbolic plane.

A new model uses complex quaternions for hyperbolic 3-space.

Tight embedding proves Gromov norm for quaternionic Kähler class.

Study describes moduli of quaternionic hyperbolic triples of points.

Study of quaternionic hyperbolic space bisectors and their decompositions.

New examples of hypersurfaces found in quaternionic hyperbolic 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…

Quaternionic hyperbolic groups stabilize complex subspaces if trace skew-field is commutative.

Study of quaternionic hyperbolic space subgroups deformations.

We introduce curvature-adapted foliations of complex hyperbolic space and study some of their properties. Generalized pseudo-Einstein hypersurfaces of complex hyperbolic space are classified. Analogous results for curvature-adapted hypersurfaces of quaternionic hyperbolic space are also obtained.

Characterizes specific harmonic manifolds using Laplacian eigenfunctions.

We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces of dimension greater than two. For the quaternionic hyperbolic spaces of dimension greater than two we reduce the classification problem to…

In this paper we give the characterization of Fuchsian groups acting on quaternionic hyperbolic 2-space.

Proves quaternionic analog of Cartan's theorem and counts arithmetic chains.

The study constructs minimal submanifolds in complex and quaternionic projective spaces.

The study finds criteria for discreteness in quaternionic hyperbolic space.

An explicit classification of homogeneous quaternionic Kaehler structures by real tensors is derived and we relate this to the representation-theoretic description found by Fino. We then show how the quaternionic hyperbolic space HH(n) is characterised by admitting homogeneous structures of a particularly simple type. …

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…

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

The study finds conditions for quaternionic structures on symmetric spaces.

This paper establishes inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane.

The paper classifies reversible and strongly reversible elements in quaternionic hyperbolic spaces.

In this paper, we obtain analogues of Jorgensen's inequality for non-elementary groups of isometries of quaternionic hyperbolic $n$-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…

Study counts and equidistributes rational points in quaternionic Heisenberg groups.

In this note, we study deformations of a non-uniform real hyperbolic lattice in quaternionic hyperbolic spaces. Specially we show that the representations of the fundamental group of the figure eight knot complement into PU(2,1) cannot be deformed in $PSp(2,1)$ out of PU(2,1) up to conjugacy.

We develop and study quaternionic and octonionic analogies of Cartan angular and Toledo invariants that are well known in the complex hyperbolic space. Using such invariants we study quasifuchsian deformations (including bendings) of quaternionic and octonionic hyperbolic manifolds.

Study inverse mean curvature flow in quaternionic hyperbolic space, proving flow properties and convergence.

We prove a Milnor-Wood inequality for representations of the fundamental group of a compact complex hyperbolic manifold in the group of isometries of quaternionic hyperbolic space. Of special interest is the case of equality, and its application to rigidity. We show that equality can only be achieved for totally geodes…

Lower bounds for systole growth in quaternionic hyperbolic manifolds.

Discrete subgroups of quaternionic hyperbolic isometries are proven under certain conditions.

Infinitely many hyperbolic links in lens space have isotopic lifts in 3-sphere.

Lower bounds for quaternionic hyperbolic orbifold volumes found.

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.

It is of interest to characterize algebraically the dynamical types of isometries of the complex and quaternionic hyperbolic planes. In the complex case, such a characterization is known from the work of Giraud-Goldman. In this paper, we offer an algebraic characterization of the isometries of the two-dimensional quate…

Paper constructs Hopf real hypersurfaces in complex hyperbolic space.

We classify the effective and transitive actions of a Lie group $G$ on an n-dimensional non-degenerate hyperboloid (also called real pseudo-hyperbolic space), under the assumption that $G$ is a closed, connected Lie subgroup of $SO_0(n-r,r+1)$, the connected component of the indefinite special orthogonal group. Assumin…

We derive an explicit lower bound on the radius of a ball embedded in a quaternionic hyperbolic manifold.

Pairs of elements in quaternionic hyperbolic space have zero measure of being strongly doubly reversible.

Classifies conjugation orbits of hyperbolic elements in various groups and surfaces.

Let ${\rm SL(2, \mathbb H)}$ be the group of $2 \times 2$ quaternionic matrices with Dieudonné determinant $1$. The group ${\rm SL(2, \mathbb H)}$ acts on the five dimensional hyperbolic space by isometries. We investigate extremality of Jørgensen type inequalities in ${\rm SL(2, \mathbb H)}$. Along the way, we derive …

Classifies matrices in the quaternionic hyperbolic unitary group.

We classify semi-Riemannian submersions with connected totally geodesic fibres from a real pseudo-hyperbolic space onto a semi-Riemannian manifold under the assumption that the dimension of the fibres is less than or equal to three and the metrics induced on fibres are negative definite. Also, we obtain the classificat…

We study the geometry of oriented right-angled hexagons in H^4, the hyperbolic 4-space, via Clifford numbers or quaternions. We show how to augment alternate sides of such a hexagon so that for the non-augmented sides, we can define quaternion half side-lengths whose angular parts are obtained from half the Euler angle…

The study classifies hypersurfaces in quaternionic space forms with constant principal curvatures.

In contrast to the classical twistor spaces whose fibres are 2-spheres, we introduce twistor spaces over manifolds with almost quaternionic structures of the second kind in the sense of P. Libermann whose fibres are hyperbolic planes. We discuss two natural almost complex structures on such a twistor space and their ho…

Researchers found the smallest covolume lattices in quaternionic hyperbolic groups.

In this paper we analyze and classify the totally geodesic subspaces of finite volume quaternionic hyperbolic orbifolds and their generalizations, locally symmetric orbifolds arising from irreducible lattices in Lie groups of the form $(\mathbf{Sp}_{2n}(\mathbb{R}))^q \times \prod_{i=1}^r \mathbf{Sp}(p_i,n-p_i) \times …

Classifies semisimple pairs in complex and quaternionic hyperbolic spaces.