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Dependence of the Gauss-Codazzi equations and the Ricci equation of Lorentz surfaces. (English) Zbl 1199.53117
Summary: The fundamental equations of Gauss, Codazzi and Ricci provide the conditions for local isometric embeddability. In general, the three fundamental equations are independent for surfaces in Riemannian 4-manifolds. In contrast, we prove in this article that for arbitrary Lorentz surfaces in Lorentzian Kaehler surfaces the equation of Ricci is a consequence of the equations of Gauss and Codazzi.
MSC:
53C40Global submanifolds (differential geometry)
53C50Lorentz manifolds, manifolds with indefinite metrics