[Paper Review] Spin(9) geometry of the octonionic Hopf fibration

Liviu Ornea, Maurizio Parton+2 more|arXiv (Cornell University)|Aug 4, 2012

Geometric Analysis and Curvature Flows20 references4 citations

TL;DR

This paper investigates the Spin(9) geometry of the octonionic Hopf fibration $S^{15} \to S^8$, proving that any vertical vector field on $S^{15}$ must have at least one zero, thereby re-proving the non-existence of $S^1$-subfibrations. It further classifies compact locally conformally parallel Spin(9)-manifolds, showing they are finitely covered by $S^{15} \times \mathbb{R}$, and provides a complete list of such manifolds via finite subgroups of $\mathrm{Sp}(1)_\Delta \subset \mathrm{Spin}(9)$.

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.

Motivation & Objective

• To analyze the Riemannian and geometric properties of the octonionic Hopf fibration $S^{15} \to S^8$ using its $\mathrm{Spin}(9)$ symmetry.
• To re-prove the non-existence of $S^1$-subfibrations by showing every vertical vector field on $S^{15}$ has at least one zero.
• To complete the general scheme of locally conformally parallel $G$-structures by treating the case $G = \mathrm{Spin}(9)$.
• To classify compact locally conformally parallel $\mathrm{Spin}(9)$-manifolds and provide explicit examples via finite subgroups of $\mathrm{Sp}(1)_\Delta \subset \mathrm{Spin}(9)$.

Proposed method

• Utilizes the action of $\mathrm{Spin}(9)$ on $S^{15}$, derived from triality in $\mathrm{SO}(8)$, to analyze the geometry of the octonionic Hopf fibration.
• Applies the Triality Principle to construct a diagonal subgroup $\mathrm{Sp}(1)_\Delta \subset \mathrm{Spin}(9)$ acting without fixed points on $S^{15}$.
• Analyzes the Lee form and Lee vector field in the context of locally conformally parallel $\mathrm{Spin}(9)$-structures, showing the Lee vector field never vanishes.
• Constructs a canonical 8-dimensional foliation on compact locally conformally parallel $\mathrm{Spin}(9)$-manifolds.
• Classifies such manifolds by identifying finite subgroups of $\mathrm{Sp}(1)_\Delta$ that act without fixed points on $S^{15}$, using known classifications of finite subgroups of $\mathrm{Sp}(1)$.
• Establishes that the universal cover of a compact locally conformally parallel $\mathrm{Spin}(9)$-manifold is conformally equivalent to $\mathbb{R}^{16} \setminus \{0\}$, the cone over $S^{15}$.

Experimental results

Research questions

• RQ1 Does every vertical vector field on $S^{15}$ tangent to the fibers of the octonionic Hopf fibration $S^{15} \to S^8$ necessarily have a zero?
• RQ2 Can the non-existence of $S^1$-subfibrations in the octonionic Hopf fibration be re-proven via vector field analysis on $S^{15}$?
• RQ3 What is the structure of compact manifolds admitting a locally conformally parallel $\mathrm{Spin}(9)$-structure?
• RQ4 How can one classify all such compact manifolds using finite group actions on $S^{15}$?
• RQ5 Which finite subgroups of $\mathrm{Sp}(1)_\Delta \subset \mathrm{Spin}(9)$ act without fixed points on $S^{15}$, and what are the corresponding $\mathrm{Spin}(9)$-manifolds?

Key findings

• Any global vector field on $S^{15}$ tangent to the fibers of the octonionic Hopf fibration $S^{15} \to S^8$ must have at least one zero, confirming the non-existence of $S^1$-subfibrations.
• The Riemannian universal covering of a compact locally conformally parallel $\mathrm{Spin}(9)$-manifold is conformally equivalent to $\mathbb{R}^{16} \setminus \{0\}$, the cone over $S^{15}$.
• Such manifolds are finitely covered by $S^{15} \times \mathbb{R}$, and admit a canonical 8-dimensional foliation.
• The classification of compact locally conformally parallel $\mathrm{Spin}(9)$-manifolds reduces to finding finite subgroups of $\mathrm{Sp}(1)_\Delta \subset \mathrm{Spin}(9)$ that act without fixed points on $S^{15}$.
• All such manifolds arise from finite subgroups of $\mathrm{Sp}(1)$—cyclic, binary dihedral, tetrahedral, octahedral, and icosahedral—lifted to $\mathrm{Sp}(1)_\Delta$, with explicit generators provided.
• The Lee vector field on any such manifold is never vanishing, implying the existence of a $\mathrm{Spin}(7)_\Delta$-structure, and reducing the isometry classification to fixed-point-free actions on $S^{15}$.