The Dirac operator on space forms of positive curvature
The Dirac operator is an elliptic differential operator naturally associated to a Riemannian spin manifold. There are only few manifolds for which one can explicitly compute the Dirac spectrum. In the present paper Killing spinors are used to determine the Dirac eigenvalues of spheres with constant curvature. They take the simple form \(\lambda = \pm ({1\over 2} \cdot\) dimension \(+ k)\), \(k \in \mathbb{N}_0\). Then the Dirac spectrum of quotients of the sphere (e.g. real hyperbolic spaces) is encoded into a Poincaré series. This is used to characterize space forms with many Killing spinors generalizing a result by A. Franc and to construct Dirac isospectral non-isometric space forms. They have metacyclic fundamental groups.
scientific article; zbMATH DE number 894114