We consider a closed hypersurface with identically zero Gauß–Kronecker curvature. We prove that if has constant mean curvature H, then is minimal, i.e., . This result extends Ramanathan's classification (Math. Z. 205 (1990) 645–658) result of closed minimal hypersurfaces of with vanishing Gauß–Kronecker curvature.
Nous considérons une hypersurface fermée (compacte et sans bord) à courbure de Gauß–Kronecker identiquement nulle. Nous prouvons que si la courbure moyenne H de est constante, alors l'hypersurface est necéssairement minimale, c.à.d, . Ce résultat généralise celui obtenu dans l'article de Ramanathan (Math. Z. 205 (1990) 645–658) concernant les hypersurfaces fermées minimales à courbure de Gauß–Kronecker identiquement nulle dans .
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Tsasa Lusala 1; André Gomes de Oliveira 1
@article{CRMATH_2005__340_6_437_0, author = {Tsasa Lusala and Andr\'e Gomes de Oliveira}, title = {Closed hypersurfaces of $ {\mathbb{S}}^{4}(1)$ with constant mean curvature and zero {Gau{\ss}{\textendash}Kronecker} curvature}, journal = {Comptes Rendus. Math\'ematique}, pages = {437--440}, publisher = {Elsevier}, volume = {340}, number = {6}, year = {2005}, doi = {10.1016/j.crma.2005.01.005}, language = {en}, }
TY - JOUR AU - Tsasa Lusala AU - André Gomes de Oliveira TI - Closed hypersurfaces of $ {\mathbb{S}}^{4}(1)$ with constant mean curvature and zero Gauß–Kronecker curvature JO - Comptes Rendus. Mathématique PY - 2005 SP - 437 EP - 440 VL - 340 IS - 6 PB - Elsevier DO - 10.1016/j.crma.2005.01.005 LA - en ID - CRMATH_2005__340_6_437_0 ER -
%0 Journal Article %A Tsasa Lusala %A André Gomes de Oliveira %T Closed hypersurfaces of $ {\mathbb{S}}^{4}(1)$ with constant mean curvature and zero Gauß–Kronecker curvature %J Comptes Rendus. Mathématique %D 2005 %P 437-440 %V 340 %N 6 %I Elsevier %R 10.1016/j.crma.2005.01.005 %G en %F CRMATH_2005__340_6_437_0
Tsasa Lusala; André Gomes de Oliveira. Closed hypersurfaces of $ {\mathbb{S}}^{4}(1)$ with constant mean curvature and zero Gauß–Kronecker curvature. Comptes Rendus. Mathématique, Volume 340 (2005) no. 6, pp. 437-440. doi : 10.1016/j.crma.2005.01.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.01.005/
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