We prove a version of Shimizu’s lemma for quaternionic hyperbolic space. Namely, consider groups of quaternionic hyperbolic isometries containing a parabolic map fixing infinity. We show that any element of such a group not fixing infinity has an isometric sphere whose radius is bounded by a function of the parabolic translation length at its centre. Mathematics Subject Classifications (2000): ...

It is shown that there is no continuous map from the quaternionic Grassmannian Grk,n(k ≥ 2, n ≥ k + 2) to the quaternionic projective space HP ∞ such that the pullback of the first Pontryagin class of the tautological bundle over HP is equal to the first Pontryagin class of the tautological bundle over Grk,n. In fact some more precise statement is proved.

We find a principle of harmonic analyticity underlying the quaternionic (quaternionKähler) geometry and solve the differential constraints which define this geometry. To this end the original 4n-dimensional quaternionic manifold is extended to a biharmonic space. The latter includes additional harmonic coordinates associated with both the tangent local Sp(1) group and an extra rigid SU(2) group...

Taking the complex nature of quantum mechanics which we observe today as a low energy effect of a broken quaternionic theory we explore the possibility that dark matter arises as a consequence of this underlying quaternionic structure to our universe. We introduce a low energy, effective, Lagrangian which incorporates the remnants of a local quaternionic algebra, investigate the stellar product...

A 7-manifold with G2 holonomy can be constructed as a R 3 bundle over a quaternionic space. We consider a quaternionic base space which is singular and its metric depends on three parameters, where one of them corresponds to an interpolation between S and CP or its non-compact analogs. This 4-d Einstein space has four isometries and the fixed point set of a generic Killing vector is discussed. ...

Abstract: In this paper, we obtain analogues of Jørgensen’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 [9], Markham [14] and Cao [2]. These results also apply to complex hyperbolic space and give improvements on results of Jiang, Kamiya ...