### A characterization of isoparametric hypersurfaces of Clifford type.

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Our concern is with the group of conformal transformations of a finite-dimensional real quadratic space of signature (p,q), that is one that is isomorphic to ${\mathbb{R}}^{p,q}$, the real vector space ${\mathbb{R}}^{p+q}$, furnished with the quadratic form ${x}^{\left(2\right)}=x\xb7x=-{x}_{1}^{2}-{x}_{2}^{2}-...-{x}_{p}^{2}+{x}_{p+1}^{2}+...+{x}_{p+q}^{2}$, and especially with a description of this group that involves Clifford algebras.

In this article fibrations of associative algebras on smooth manifolds are investigated. Sections of these fibrations are spinor, co spinor and vector fields with respect to a gauge group. Invariant differentiations are constructed and curvature and torsion of invariant differentiations are calculated.

The concept of supercomplex structure is introduced in the pseudo-Euclidean Hurwitz pairs and its basic algebraic and geometric properties are described, e.g. a necessary and sufficient condition for the existence of such a structure is found.

It is a common belief among theoretical physicists that the charge conjugation of the Dirac equation has an analogy in higher dimensional space-times so that in an 8-dimensional space-time there would also be Maiorana spinors as eigenspinors of a charge conjugation, which would swap the sign of the electric charge of the Dirac equation. This article shows that this mistaken belief is based on inadequate distinction between two kinds of charge conjugation: the electric conjugation swapping the sign...

In the Fourier theory of functions of one variable, it is common to extend a function and its Fourier transform holomorphically to domains in the complex plane C, and to use the power of complex function theory. This depends on first extending the exponential function eixξ of the real variables x and ξ to a function eizζ which depends holomorphically on both the complex variables z and ζ .Our thesis is this. The natural analog in higher dimensions is to extend a function of m real variables monogenically...

In this paper we show that well-known relationships connecting the Clifford algebra on negative euclidean space, Vahlen matrices, and Möbius transformations extend to connections with the Möbius loop or gyrogroup on the open unit ball $B$ in $n$-dimensional euclidean space ${\mathbb{R}}^{n}$. One notable achievement is a compact, convenient formula for the Möbius loop operation $a*b=(a+b){(1-ab)}^{-1}$, where the operations on the right are those arising from the Clifford algebra (a formula comparable to $(w+z){(1+\overline{w}z)}^{-1}$ for the Möbius loop multiplication...