On the stability of the quaternion projective space

TL;DR

This paper proves that quaternion space forms with negative curvature are stable, while quaternion projective spaces are unstable, by analyzing the index of the identity map in relation to curvature.

Contribution

It establishes the stability of compact quaternion space forms with negative curvature and highlights the instability of quaternion projective spaces, extending understanding of their geometric properties.

Findings

Index of the identity map is zero for quaternion space forms with negative curvature.

Negative curvature quaternion space forms are stable.

Quaternion projective spaces are unstable.

Abstract

The aim of this note is to prove that the index of the identity map on a quaternion space form of constant quaternion sectional curvature $c$ is zero, provided that $c<0$. As an immediate consequence, it is established that any compact quaternion space form of negative quaternion sectional curvature is stable and it is emphasized that, on the contrary (but in agreement with some known results), any quaternion projective space is unstable.