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HYPERSURFACES OF ${\mathbb S}^{n+1}$ WITH TWO DISTINCT PRINCIPAL CURVATURES

Published online by Cambridge University Press:  31 January 2005

JOSÉ N. BARBOSA
Affiliation:
Universidade Federal da Bahia, Instituto de Matemática, Departamento de Matemática, Av. Ademar de Barros, s/n, Campus de Ondina – Ondina, Salvador, BA, 40.170-110, Brazil e-mail: jnelson@ufba.br
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Abstract

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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 $S^{n-1}(r) \times S^1(\sqrt{1-r^2})$.

Type
Research Article
Copyright
2005 Glasgow Mathematical Journal Trust