G-invariant Spin Structures on Spheres

TL;DR

This paper classifies which Lie groups acting transitively on spheres preserve the sphere's unique spin structure, using differential and representation theoretic methods.

Contribution

It provides a complete classification of Lie group actions on spheres that leave the spin structure invariant, employing two distinct analytical approaches.

Findings

Identifies specific Lie groups preserving the spin structure on spheres.

Provides two independent proofs of the classification.

Enhances understanding of symmetry and spin geometry on spheres.

Abstract

We examine which of the compact connected Lie groups that act transitively on spheres of different dimensions leave the unique spin structure of the sphere invariant. We study the notion of invariance of a spin structure and prove this classification in two different ways; through examining the differential of the actions and through representation theory.