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Pseudo-spherical submanifolds with 1-type pseudo-spherical Gauss map. (English) Zbl 1377.53073

Summary: In this work, we study pseudo-Riemannian submanifolds of a pseudo-sphere with 1-type pseudo-spherical Gauss map. First, we classify Lorentzian surfaces in a 4-dimensional pseudo-sphere \(\mathbb S^4_s(1)\) with index \(s\), \(s=1,2\), and having harmonic pseudo-spherical Gauss map. Then we give a characterization theorem for pseudo-Riemannian submanifolds of a pseudo-sphere \(\mathbb S^{m-1}_s(1)\subset\mathbb E^m_s\) with 1-type pseudo-spherical Gauss map, and we classify spacelike surfaces and Lorentzian surfaces in the de Sitter space \(\mathbb S^4_1(1)\subset\mathbb E^5_1\) with 1-type pseudo-spherical Gauss map. Finally, according to the causal character of the mean curvature vector we obtain the classification of submanifolds of a pseudo-sphere having 1-type pseudo-spherical Gauss map with nonzero constant component in its spectral decomposition.

MSC:

53C40 Global submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53B25 Local submanifolds
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References:

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