Generalized Killing spinors associated with the Ricci tensor

Abstract

In this paper, we introduce the notion of Ricci Killing spinors on Riemannian spin manifolds, which form a class between generalized Killing spinors and standard Killing spinors. We establish a rigidity result for Ricci Killing spinors and characterize simply connected Sasakian $η$-Einstein spin manifolds admitting Ricci Killing spinors. We also give a complete classification of Ricci Killing spinors on simply connected $3$-dimensional unimodular Lie groups with left-invariant Riemannian metrics. In particular, we obtain new examples of generalized Killing spinors which are not Killing spinors. Finally, for the non-Killing Ricci Killing spinors obtained in the Lie group case, we explicitly determine the corresponding Ricci-flat ambient manifolds carrying parallel spinors.