The Role of Spin(9) in Octonionic Geometry

TL;DR

This paper reviews the interactions between Spin(9) symmetry and octonionic geometries, exploring explicit forms, classical vector field problems, and applications to special manifolds and Grassmannians.

Contribution

It provides a comprehensive overview of Spin(9) in octonionic geometry, including explicit forms, classical problems, and applications to special manifolds and algebraic structures.

Findings

Explicit description of the Spin(9) canonical 8-form.

Analysis of Spin(9) in vector fields on spheres.

Applications to Cayley-Rosenfeld planes and Grassmannians.

Abstract

Starting from the 2001 Thomas Friedrich's work on Spin(9), we review some 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 as well as 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.