Generalized cylinders in semi-Riemannian and spin geometry
Copyright policy )Generalized cylinders in semi-Riemannian and spin geometry
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.
29 pages, 2 figures
- École Polytechnique France
- University of Potsdam Germany
Mathematics - Differential Geometry, energy-momentum tensor of a spinor, Local differential geometry of Lorentz metrics, indefinite metrics, generalized Killing spinors, Dirac operator, Spin and Spin\({}^c\) geometry, FOS: Physical sciences, fundamental theorem of hypersurface theory, 510, variation formula for Dirac operator, identification of spinors, FOS: Mathematics, hypersurface theory, Codazzi tensors, generalized cylinder, Mathematical Physics, space of Lorentzian metrics, Institut für Mathematik, Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces, Mathematical Physics (math-ph), 53C27,53A07,53B30, Differential Geometry (math.DG), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], 53C27, 53A07, 53B30
Mathematics - Differential Geometry, energy-momentum tensor of a spinor, Local differential geometry of Lorentz metrics, indefinite metrics, generalized Killing spinors, Dirac operator, Spin and Spin\({}^c\) geometry, FOS: Physical sciences, fundamental theorem of hypersurface theory, 510, variation formula for Dirac operator, identification of spinors, FOS: Mathematics, hypersurface theory, Codazzi tensors, generalized cylinder, Mathematical Physics, space of Lorentzian metrics, Institut für Mathematik, Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces, Mathematical Physics (math-ph), 53C27,53A07,53B30, Differential Geometry (math.DG), [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG], 53C27, 53A07, 53B30
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