Quaternionic Killing spinors (Q1386173)

Let \(M\) be a closed quaternionic Kähler manifold. Suppose \(M\) is spin and of positive scalar curvature. It was shown by the same three authors in [`Eigenvalue estimates for the Dirac operator on quaternionic Kähler manifolds' (Preprint 507, SFB 256, Bonn) (1997; dg-ga/9709014)] that all eigenvalues \(\lambda\) of the Dirac operator satisfy \[ \lambda^2 \geq {\kappa\over 4}{n+3\over n+2} \] where \(n\) is the quaternionic dimension of \(M\) and \(\kappa\) its scalar curvature. Note that \(\kappa\) is constant because all quaternionic Kähler manifolds are Einstein. This estimate is sharp since equality is attained for the smallest eigenvalue of quaternionic projective space. It is natural to ask if there are quaternionic Kähler manifolds other than projective space for which this estimate is sharp. The partial answer given in the present paper is: not among symmetric quaternionic Kähler manifolds. In a more recent paper [Commun. Math. Phys. 199, 327-349 (1998)], the authors improve this result and remove the restriction to symmetric spaces. The answer to the question is simply no. The proof in the present article is based on a quaternionic version of the Killing spinor equation and the classification of symmetric spaces.