Differential Geometry
A Note on compact hypersurfaces in a Euclidean space
Comptes Rendus. Mathématique, Volume 350 (2012) no. 21-22, pp. 971-974.

In this Note, we show that the presence of a unit Killing vector field ξ on an orientable compact hypersurface of a Euclidean space with shape operator A and induced metric g such that g(Aξ,ξ) is a constant, renders it to be a round sphere and also influences the dimension of the ambient Euclidean space.

Supposons la donnée dʼun champs de vecteurs unitaire de Killing ξ sur une hypersurface compacte, orientable, dʼun espace euclidien, avec opérateur de forme A et métrique induite tels que g(Aξ,ξ) soit constant. Nous montrons dans cette Note que cela impose que lʼhypersurface est nécessairement isométrique à une sphère de courbure constante et que lʼespace euclidien ambiant est de dimension paire.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2012.10.027
Deshmukh, Sharief 1

1 Department of Mathematics, College of Science, King Saud University, P.O. Box-2455, Riyadh 11451, Saudi Arabia
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Deshmukh, Sharief. A Note on compact hypersurfaces in a Euclidean space. Comptes Rendus. Mathématique, Volume 350 (2012) no. 21-22, pp. 971-974. doi : 10.1016/j.crma.2012.10.027. http://www.numdam.org/articles/10.1016/j.crma.2012.10.027/

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