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Classification of minimal Lorentzian surfaces in $$\mathbb{S}^4_2(1)$$ with constant Gaussian and normal curvatures. (English) Zbl 1357.53026
Summary: In this paper we consider Lorentzian surfaces in the $$4$$-dimensional pseudo-Riemannian sphere $$\mathbb{S}^4_2(1)$$ with index $$2$$ and curvature one. We obtain the complete classification of minimal Lorentzian surfaces $$\mathbb{S}^4_2(1)$$ whose Gaussian and normal curvatures are constants. We conclude that such surfaces have the Gaussian curvature $$1/3$$ and the absolute value of normal curvature $$2/3$$. We also give some explicit examples.

MSC:
 53B25 Local submanifolds 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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