Generalized cylinders in semi-Riemannian and spin geometry

2003/03/07 by Christian Baer, Christian B�r, Paul Gauduchon +1 · 150 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Constant (computer programming) #Constant curvature #Curvature #Dirac operator #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Hypersurface #Manifold (fluid mechanics) #Mathematical analysis #Mathematical physics #Mathematics #Pure mathematics #Ricci-flat manifold #Scalar curvature #Sectional curvature #Space (punctuation) #Spin (aerodynamics) #Spinor #math-ph #math.DG #math.MP #msc:53A07 #msc:53B30 #msc:53C27

paper · pdf · doi:10.1007/s00209-004-0718-0

published in Mathematische Zeitschrift 249(3), 545-580 (Springer Science+Business Media) · 29 pages, 2 figures

arxiv created 2003/03/07 · openalex publication_date 2005/01/07 · openalex created_date 2016/06/24 · arxiv updated 2019/01/08 · openalex updated_date 2026/08/05

Abstract

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.

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