Killing spinors, twistor - spinors and Hijazi inequality

✍ Scribed by Andre Lichnerowicz

• Publisher: Elsevier Science
• Year: 1988
• Tongue: English
• Weight: 772 KB
• Volume: 5
• Category: Article
• ISSN: 0393-0440
• DOI: 10.1016/0393-0440(88)90011-3

✦ Synopsis

Let 1W, g) be a spin manifold of dimension n. In terms of the Dirac operator P of (W, g), we introduce on the spinor fields a conformally covariant first-order operator D that is strictly connected with the twisror-spinors. We show that the operator (L~-p) (p (n/4(n .-1))R) is positive. For a compact spin manifold of dimension n ~3, the existences of harmonic spinors and iwistorspinors ~0 are mutually exclusive, except for the parallel spinors. By means of a universal formula, we show that the Hijazi inequality [8] holds for every spinor field such that (Pi,I', P~li)= X 2(~,1i,~i) (X = const). In the limiting case, the manifold admits a Killing spinor wich can be evaluated in terms of ~ti. Using the Yamabe-Schoen theorem [15], we prove that, if the space r of the twistor-spinors of (W, g) is not reduced to zero, there is a conformal change of the metric g giving a manifold with Killing spinors * 0. Interpretation of dim ii' in terms of these spaces of Killing spinors. If the compact spin manifold (W, g) of dimension n ~3 is not conformally isometric with the sphere, every twistor-spinor is without zero on W.