Related papers: The Role of Spin(9) in Octonionic Geometry
We deal with Riemannian properties of the octonionic Hopf fibration S^{15}-->S^8, in terms of the structure given by its symmetry group Spin(9). In particular, we show that any vertical vector field has at least one zero, thus reproving the…
An explicit expression of the canonical 8-form on a Riemannian manifold with a Spin(9)-structure, in terms of the nine local symmetric involutions involved, is given. The list of explicit expressions of all the canonical forms related to…
It is well known that there is a unique $Spin(9)$-invariant 8-form on the octonionic plane that naturally yields a canonical differential 8-form on any Riemannian manifold with a weak $Spin(9)$-structure. Over the decades, this invariant…
We study Spin(9)-structures on 16-dimensional Riemannian manifolds and characterize the geometric types admitting a connection with totally skew-symmetric torsion.
We describe explicitly the algebra of Spin(9)-invariant, translation-invariant, continuous valuations on the octonionic plane. Namely, we present a basis in terms of invariant differential forms and determine the Bernig-Fu convolution on…
For a Spin(9)-structure on a Riemannian manifold M^16 we write explicitly the matrix psi of its K\"ahler 2-forms and the canonical 8-form Phi. We then prove that Phi coincides up to a constant with the fourth coefficient of the…
The dimensions of the spaces of $k$-homogeneous $\mathrm{Spin}(9)$-invariant valuations on the octonionic plane are computed using results from the theory of differential forms on contact manifolds as well as octonionic geometry and…
Quaternionic and octonionic realizations of Clifford algebras and spinors are classified and explicitly constructed in terms of recursive formulas. The most general free dynamics in arbitrary signature space-times for both quaternionic and…
In this paper we revisit the concept of conformality in the sense of Gauss in the context of octonions and Clifford algebras. We extend a characterization of conformality in terms of a system of partial differential equations and…
This paper is meant to be an informative introduction to spinor representations of Clifford algebras. In this paper we will have a look at Clifford algebras and the octonion algebra. We begin the paper looking at the quaternion algebra…
The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the non-associativity and non-commutativity of this division algebra…
In this review, basic definitions of spin geometry are given and some of its applications to supersymmetry, supergravity and condensed matter physics are summarized. Clifford algebras and spinors are defined and the first-order differential…
In a recent paper, algebraic descriptions for all non-relativistic spins were derived by elementary means directly from the Lie algebra $\specialorthogonalliealgebra{3}$, and a connection between spin and the geometry of Euclidean…
We give an inductive construction for irreducible Clifford systems on Euclidean vector spaces. We then discuss how this notion can be adapted to Riemannian manifolds, and outline some developments in octonionic geometry.
A comparison among different constructions of the quaternionic $4$-form $\Phi_{Sp(2)Sp(1)}$ and of the Cayley calibration $\Phi_{Spin(7)}$ shows that one can start for them from the same collections of "K\"ahler 2-forms", entering in…
This paper investigates the geometric structures and properties of 8-dimensional manifolds with Spin(7)-holonomy. We focus on the characterization and implications of 4-planes within these manifolds, which are endowed with an almost…
The aim of the present paper is the investigation of $Spin(9)$-structures on 16-dimensional manifolds from the point of view of topology as well as holonomy theory. First we construct several examples. Then we study the necessary…
We interpret an open orbit in a 32-dimensional representation space of Spin(9,1) x SL(2,R) as a substitute for the non-existent group of invertible 2x2 matrices over the octonions and study various natural homogeneous subspaces. The…
Introducing products between multivectors of Cl(0,7) and octonions, resulting in an octonion, and leading to the non-associative standard octonionic product in a particular case, we generalize the octonionic X-product, associated with the…
By constructing concrete complex-oriented maps we show that the eight-fold of the generator of the third integral cohomology of the spin groups Spin(7) and Spin(8) is in the image of the Thom morphism from complex cobordism to singular…