[Sur les hypersurfaces compactes dʼun espace euclidien]
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 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.
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 is a constant, renders it to be a round sphere and also influences the dimension of the ambient Euclidean space.
Accepté le :
Publié le :
Sharief Deshmukh 1
@article{CRMATH_2012__350_21-22_971_0, author = {Sharief Deshmukh}, title = {A {Note} on compact hypersurfaces in a {Euclidean} space}, journal = {Comptes Rendus. Math\'ematique}, pages = {971--974}, publisher = {Elsevier}, volume = {350}, number = {21-22}, year = {2012}, doi = {10.1016/j.crma.2012.10.027}, language = {en}, }
Sharief Deshmukh. 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. https://comptes-rendus.academie-sciences.fr/mathematique/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.
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