Generalized Killing spinors on spheres

We study generalized Killing spinors on round spheres Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^n$$\end{document}. We show that on the standard sphere S8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^8$$\end{document} any generalized Killing spinor has to be an ordinary Killing spinor. Moreover, we classify generalized Killing spinors on Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^n$$\end{document} whose associated symmetric endomorphism has at most two eigenvalues and recover in particular Agricola–Friedrich’s canonical spinor on 3-Sasakian manifolds of dimension 7. Finally, we show that it is not possible to deform Killing spinors on standard spheres into genuine generalized Killing spinors.