Preprint Review Version 1 This version is not peer-reviewed

The Role of Spin(9) in Octonionic Geometry

Version 1 : Received: 20 September 2018 / Approved: 21 September 2018 / Online: 21 September 2018 (06:07:27 CEST)

A peer-reviewed article of this Preprint also exists.

Parton, M.; Piccinni, P. The Role of Spin(9) in Octonionic Geometry. Axioms 2018, 7, 72. Parton, M.; Piccinni, P. The Role of Spin(9) in Octonionic Geometry. Axioms 2018, 7, 72.

Journal reference: Axioms 2018, 7, 72
DOI: 10.3390/axioms7040072

Abstract

Starting from Thomas Friedrich’s work “Weak Spin(9) structures on 16-dimensional Riemannian manifolds”, we review several interactions between Spin(9) and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the Spin(9) canonical 8-form and its analogies with quaternionic geometry, the role of Spin(9) both in the classical problems of vector fields on spheres and in the geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel Spin(9) manifolds in the framework of intrinsic torsion. Finally, we discuss applications of Clifford systems and Clifford structures to Cayley-Rosenfeld planes and to three series of Grassmannians.

Subject Areas

Spin(9); octonions; vector fields on spheres; Hopf fibration; locally conformally parallel; Clifford structure; Clifford system; symmetric spaces

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