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Spin manifolds, Killing spinors and universality of the Hijazi inequality. (English) Zbl 0624.53034

About twenty-five years ago the author initiated an ambitious and elegant program of reformulating the quantum theory of fields on a curved Riemannian manifold of hyperbolic normal signature. Although this program was intended to provide a framework for ultimately quantizing general relativity, it also involved the study of spin-structures, spin- manifolds, the Dirac operator, harmonic spinors, and a number of other objects of mathematical interest. In the present paper, prompted by the more recent work of T. Friedrich [Math. Nachr. 97, 117-146 (1980; Zbl 0462.53027)] and O. Hijazi [Commun. Math. Phys. 104, 151-162 (1986; Zbl 0593.58040)], the author resumes this study. Contents include: an introduction, spin manifolds and corresponding connections, a universal formula, Killing spinors, parallel forms, isometries, conformal change of metric, and a generalization of the Hijazi inequality.
Reviewer: J.D.Zund

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C27 Spin and Spin\({}^c\) geometry
53C80 Applications of global differential geometry to the sciences
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