Complete Riemannian manifolds with imaginary Killing spinors (Q909256)

A Killing spinor on a complete Riemannian spin manifold \(M^ n\) is a section \(\psi\in \Gamma (S)\) of the spin bundle S satisfying for any vector \(X\in T\) the differential equation \(\nabla_ X\psi =\beta \cdot X\cdot \psi,\) where \(\beta \in {\mathbb{C}}\setminus \{0\}\) is a complex number. If \(M^ n\) admits a Killing spinor then it is an Einstein space with scalar curvature \(R=4n(n-1)\beta^ 2.\) Hence the ``Killing number'' \(\beta\) is either real or imaginary. The aim of the author's paper is the construction and isometric classification of all complete Einstein spaces with imaginary Killing spinors. It turns out that they are isometric to warped products \((F^{n-1}\times R,\quad e^{-4\mu t}h\oplus dt^ 2),\quad \beta =i\mu,\) where \((F^{n-1},h)\) is a complete Riemannian spin manifold with parallel spinor. If \(\psi\) is a Killing spinor then \(q_{\psi}:=| \psi (x)|^ 4-(1/4\mu^ 2)| \text{grad} | \psi |^ 2 |^ 2\geq 0\) is constant. A complete Riemannian manifold \(M^ n\) admitting an imaginary Killing spinor \(\psi\) such that \(q_{\psi}>0\) is isometric to the hyperbolic space \(H^ n(-4\mu^ 2)\).