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Hypersurfaces with semi-parallel Blaschke tensor in the unit sphere (I)
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In this paper, focusing on Möbius geometry of submanifolds, we initiate the study of umbilic-free hypersurfaces in the unit sphere [Formula: see text] with semi-parallel Blaschke tensor. As the results, relating to the Möbius second fundamental form [Formula: see text], we first show that for such hypersurfaces the Möbius form is closed and the Blaschke tensor has at most three distinct eigenvalues provided [Formula: see text]. Then, for [Formula: see text], we classify such hypersurfaces under the additional condition of either [Formula: see text], or [Formula: see text], or their Blaschke tensor has three distinct eigenvalues. Finally, we establish a complete classification for [Formula: see text]-dimensional hypersurfaces of [Formula: see text] with semi-parallel Blaschke tensor.
World Scientific Pub Co Pte Ltd
Title: Hypersurfaces with semi-parallel Blaschke tensor in the unit sphere (I)
Description:
In this paper, focusing on Möbius geometry of submanifolds, we initiate the study of umbilic-free hypersurfaces in the unit sphere [Formula: see text] with semi-parallel Blaschke tensor.
As the results, relating to the Möbius second fundamental form [Formula: see text], we first show that for such hypersurfaces the Möbius form is closed and the Blaschke tensor has at most three distinct eigenvalues provided [Formula: see text].
Then, for [Formula: see text], we classify such hypersurfaces under the additional condition of either [Formula: see text], or [Formula: see text], or their Blaschke tensor has three distinct eigenvalues.
Finally, we establish a complete classification for [Formula: see text]-dimensional hypersurfaces of [Formula: see text] with semi-parallel Blaschke tensor.
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