Linear perturbations of metrics with holonomy Spin(7)

TL;DR

This paper studies linear perturbations of Spin(7)-structures, revealing that only specific nilpotent matrix perturbations are nontrivial, and shows that certain invariant perturbations of a Bryant-Salamon metric are isometric.

Contribution

It characterizes the form of linear perturbations of Spin(7)-structures and demonstrates the isometry of metrics obtained through invariant perturbations.

Findings

Nontrivial perturbations are determined by rank one nilpotent matrices.

Invariant perturbations of the Bryant-Salamon metric are isometric.

The method clarifies the structure of perturbations preserving Spin(7) holonomy.

Abstract

We apply the method of linear perturbations to the case of Spin(7)-structures, showing that the only nontrivial perturbations are those determined by a rank one nilpotent matrix. We consider linear perturbations of the Bryant-Salamon metric on the spin bundle over $S^4$ that retain invariance under the action of Sp(2), showing that the metrics obtained in this way are isometric.