Some remarks on the Hijazi inequality and generalizations of the Killing equation for spinors

We generalize the well-known lower estimates for the first eigenvalue of th e Dirac operator on a compact Riemannian spin manifold proved by Friedrich [Math. Nachr. 97 (1980) 117-146] and Hijazi [Math. Phys. 104(1986) 151-162; J. Geom, Phys. 16 (1995) 27-38], The special solutions of the Einstein-Dir ac equation constructed recently by Friedrich/Kim are examples for the limi ting case of these inequalities, The discussion of the limiting case of the se estimates yields two new field equations generalizing the Killing equati on as well as the weak Killing equation for spinor fields. Finally, we disc uss the two-and three-dimensional case in more detail. (C) 2001 Elsevier Sc ience B.V. All rights reserved. MSC: 53C25; 58G30.