A Note on the Construction of Complex and Quaternionic Vector Fields on Spheres

TL;DR

This paper explores the relationship between real, complex, and quaternionic vector fields on spheres, revealing a specific formula connecting their numbers of linearly independent fields based on sphere dimension.

Contribution

It establishes a precise relationship between the counts of complex and quaternionic vector fields on spheres, extending understanding of their algebraic and topological structures.

Findings

Number of complex vector fields on (4n-1)-sphere is twice the quaternionic count plus d.

The value of d is either 1 or 3, depending on the context.

Provides a formula linking different types of vector fields on spheres.

Abstract

A relationship between real, complex, and quaternionic vector fields on spheres is given by using a relationship between the corresponding standard inner products. The number of linearly independent complex vector fields on the standard $(4n-1)$-sphere is shown to be twice the number of linearly independent quaternionic vector fields plus $d$, where $ d = 1 \mbox{ or } 3$.