We discuss the differing definitions of complex and quaternionic projective group representations employed by us and by Emch. The definition of Emch (termed here a strong projective representation) is too restrictive to accommodate quaternionic Hilbert space embeddings of complex projective representations. Our definition (termed here a weak projective representation) encompasses such embedding...

We discuss the differing definitions of complex and quaternionic projective group representations employed by us and by Emch. The definition of Emch (termed here a strong projective representation) is too restrictive to accommodate quaternionic Hilbert space embeddings of complex projective representations. Our definition (termed here a weak projective representation) encompasses such embedding...

We prove that a compact quaternionic-Kähler manifold of dimension 4n ≥ 8 admitting a conformal-Killing 2-form which is not Killing, is isomorphic to the quaternionic projective space, with its standard quaternionicKähler structure.

A suitable parameterization of space-time in terms of one complex and three quaternionic imaginary units allows Lorentz transformations to be implemented as multiplication by complex-quaternionic numbers rather than matrices. Maxwell’s equations reduce to a single equation.

We study the moduli space of quaternionic Kähler structures on a compact manifold of dimension 4n ≥ 12 from a point of view of Riemannian geometry, not twistor theory. Then we obtain a rigidity theorem for quaternionic Kähler structures of nonzero scalar curvature by observing the moduli space.

In this article, we establish a sharp inequality involving δ-invariant introduced by Chen for submanifolds in quaternionic space forms of constant quaternionic sectional curvature with arbitrary codimension. Mathematics Subject Classification: 53B25, 53B35.

We present a geometric construction of a new class of hyper-Kähler manifolds with torsion. This involves the superposition of the four-dimensional hyper-Kähler geometry with torsion associated with the NS-5-brane along quaternionic planes in H k. We find the moduli space of these geometries and show that it can be constructed using the bundle space of the canonical quaternionic line bundle over...

Spaces in the genus of infinite quaternionic projective space which admit essential maps from infinite complex projective space are classified. In these cases the sets of homotopy classes of maps are described explicitly. These results strengthen the classical theorem of McGibbon and Rector on maximal torus admissibility for spaces in the genus of infinite quaternionic projective space. An inte...

We develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group. We first review the Cauchy-Fueter and Poisson formulas and explain their representation theoretic meaning. The requirement of unitarity of representations leads us to the extensions of these formulas in the Minkowski space, which can be viewed as another real form of...

We present a new construction of translation invariant continuous valuations on convex compact subsets of a quaternionic space H n ≃ R4n. This construction is based on the theory of plurisubharmonic functions of quaternionic variables started by the author in [4] and [5] which is based in turn on the notion of non-commutative determinants. In this paper we also establish some new properties of ...