Spin(9) geometry of the octonionic Hopf fibration

TL;DR

This paper explores the geometric and topological properties of the octonionic Hopf fibration using Spin(9) symmetry, establishing new results on vector fields and classifying certain Spin(9)-manifolds.

Contribution

It provides a new proof of the non-existence of S^1 subfibrations and classifies compact locally conformally parallel Spin(9)-manifolds.

Findings

Vertical vector fields have zeros, confirming no S^1 subfibrations.

Classification of compact locally conformally parallel Spin(9)-manifolds.

Examples of locally conformally parallel Spin(9)-manifolds provided.

Abstract

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 non-existence of S^1 subfibrations. We then discuss Spin(9)-structures from a conformal viewpoint and determine the structure of compact locally conformally parallel Spin(9)-manifolds. Eventually, we give a list of examples of locally conformally parallel Spin(9)-manifolds.