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A note on hypersurfaces in a sphere

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Abstract

We study the influence of a unit Killing vector field on the geometry of a hypersurface in the unit sphere. The combination of the Killing vector field on the hypersurface and the conformal vector field on the ambient sphere triggers the presence of four specific smooth functions on the hypersurface, we use these four functions to derive different sufficient conditions for a hypersurface to be the totally geodesic sphere and for a minimal hypersurface to be the totally geodesic sphere, Clifford minimal hypersurface respectively. In particular we classify compact minimal hypersurfaces with a unit Killing vector field in the unit sphere.

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Authors and Affiliations

  1. Department of Mathematics, College of Science, King Saud University, P.O. Box#2455, Riyadh, 11451, Saudi Arabia

    Sharief Deshmukh

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  1. Sharief Deshmukh
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Correspondence to Sharief Deshmukh.

Additional information

Communicated by D. V. Alekseevsky.

This Work is supported by King Saud University, Deanship of Scientific Research, Research group Project No. RGP-VPP-182.

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Deshmukh, S. A note on hypersurfaces in a sphere. Monatsh Math 174, 413–426 (2014). https://doi.org/10.1007/s00605-013-0549-3

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  • DOI: https://doi.org/10.1007/s00605-013-0549-3

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