Bär, Christian; Gauduchon, Paul; Moroianu, Andrei Generalized cylinders in semi-Riemannian and spin geometry. (English) Zbl 1068.53030 Math. Z. 249, No. 3, 545-580 (2005). Summary: We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result directly extends to semi-Riemannian manifolds and to embeddings into spaces of constant curvature. We also give a new way to identify spinors for different metrics and to derive the variation formula for the Dirac operator. Moreover, we show that generalized Killing spinors for Codazzi tensors are restrictions of parallel spinors. Finally, we study the space of Lorentzian metrics and give a criterion when two Lorentzian metrics on a manifold can be joined in a natural manner by a 1-parameter family of such metrics. Cited in 5 ReviewsCited in 76 Documents MSC: 53C27 Spin and Spin\({}^c\) geometry 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces 53B30 Local differential geometry of Lorentz metrics, indefinite metrics Keywords:hypersurface theory; Dirac operator; generalized Killing spinors; Codazzi tensors PDFBibTeX XMLCite \textit{C. Bär} et al., Math. Z. 249, No. 3, 545--580 (2005; Zbl 1068.53030) Full Text: DOI arXiv References: [1] Bär, C.: Real Killing spinors and holonomy. Commun. Math. Phys. 154, 509-521 (1993) · Zbl 0778.53037 [2] Baum, H.: Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten. B. G. Teubner Verlagsgesellschaft, Leipzig, 1981 · Zbl 0519.53054 [3] Bourguignon, J.-P., Gauduchon, P.: Spineurs, opérateurs de Dirac et variations de métriques. Commun. Math. Phys. 144, 581-599 (1992) · Zbl 0755.53009 [4] Deligne, P. et al., (eds.), Quantum fields and strings: A course for mathematicians, Vol. 1, AMS, 1999 [5] Friedrich, T., Kim, E. C.: Some remarks on the Hijazi inequality and generalizations of the Killing equation for spinors. J. Geom. Phys. 37, 1-14 (2001) · Zbl 0979.53052 [6] Kobayashi, S., Nomizu, K.: Foundations of differential geometry, Vol. 2, Interscience Publishers, John Wiley & Sons, New York, Chichester, Brisbane, Toronto, 1969 · Zbl 0175.48504 [7] Morel, B.: The energy-momentum tensor as a second fundamental form. Preprint, 2003, math.DG/0302205 [8] Mounoud, P.: Some topological and metrical properties of the space of Lorentz metrics. Diff. Geom. Appl. 15, 47-57 (2001) · Zbl 1039.53077 [9] O?Neill, B.: Semi-Riemannian geometry. Academic Press, New York, London, 1983 [10] Petersen, P.: Riemannian geometry. Springer Verlag, New York, 1998 · Zbl 0914.53001 [11] Wolf, J.: Spaces of constant curvature. Publish or Perish, Wilmington, Delaware, 1984 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.