HYPERSURFACES OF ${\mathbb S}^{n+1}$ WITH TWO DISTINCT PRINCIPAL CURVATURES
Glasgow mathematical journal, Tome 47 (2005) no. 1, pp. 149-153
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The aim of this paper is to prove that the Ricci curvature ${\rm Ric}_M$ of a complete hypersurface $M^n$, $n\,{\ge}\,3$, of the Euclidean sphere $\mathbb{S}^{n+1}$, with two distinct principal curvatures of multiplicity 1 and $n-1$, satisfies $\sup {\rm Ric}_M\,{\ge}\,\inf\, f(H)$, for a function\, $f$ depending only on $n$ and the mean curvature $H$. Supposing in addition that $M^n$ is compact, we will show that the equality occurs if and only if $H$ is constant and $M^n$ is isometric to a Clifford torus $S^{n-1}(r) \times S^1(\sqrt{1-r^2})$.
BARBOSA, JOSÉ N. HYPERSURFACES OF ${\mathbb S}^{n+1}$ WITH TWO DISTINCT PRINCIPAL CURVATURES. Glasgow mathematical journal, Tome 47 (2005) no. 1, pp. 149-153. doi: 10.1017/S0017089504002137
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author = {BARBOSA, JOS\'E N.},
title = {HYPERSURFACES {OF} ${\mathbb S}^{n+1}$ {WITH} {TWO} {DISTINCT} {PRINCIPAL} {CURVATURES}},
journal = {Glasgow mathematical journal},
pages = {149--153},
year = {2005},
volume = {47},
number = {1},
doi = {10.1017/S0017089504002137},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089504002137/}
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