On the topology and the geometry of SO(3)-manifolds

TL;DR

This paper investigates the topological and geometric properties of 5-dimensional manifolds that admit a specific SO(3) structure derived from a 5-dimensional irreducible representation, exploring their differential geometry and topology.

Contribution

It introduces and analyzes the topology and differential geometry of 5-manifolds with a nonstandard SO(3) embedding, expanding understanding of their structure and properties.

Findings

Characterization of the topology of SO(3)_ ext{ir}-structured manifolds

Analysis of the geometric properties induced by the SO(3)_ ext{ir} structure

Insights into the differential geometry of these special 5-manifolds

Abstract

Consider the nonstandard embedding of SO(3) into SO(5) given by the 5-dimensional irreducible representation of SO(3), henceforth called SO(3)_\ir. In this note, we study the topology and the differential geometry of 5-dimensional Riemannian manifolds carrying such an SO(3)_\ir structure, i.\,e. with a reduction of the frame bundle to SO(3)_\ir.