Higher rank homogeneous Clifford structures

TL;DR

This paper establishes upper bounds for the rank of homogeneous Clifford structures on compact manifolds with non-zero Euler characteristic, identifying exact limits and unique solutions in key cases.

Contribution

It provides new upper bounds for the rank of homogeneous Clifford structures and characterizes the limiting cases with unique solutions.

Findings

Upper bounds for rank r depending on parameter a

Exact solutions identified in limiting cases

Four specific limiting cases described

Abstract

We give an upper bound for the rank $r$ of homogeneous (even) Clifford structures on compact manifolds of non-vanishing Euler characteristic. More precisely, we show that if $r=2^a\cdot b$ with $b$ odd, then $r\le 9$ for $a=0$, $r\le 10$ for $a=1$, $r\le 12$ for $a=2$ and $r\le 16$ for $a\ge 3$. Moreover, we describe the four limiting cases and show that there is exactly one solution in each case.