Hypersurfaces Immersed in a Conformally Flat Riemannian Manifold : Dedicated to Professor S. Tachibana on his 50th Birthday

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  • Hypersurfaces Immersed in a Conformally

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In 1968, J. Simons [3] has established a formula for the Laplacian of the second fundamental form of a submanifold and has obtained some applications in the case of minimal hypersurfaces in the sphere. A formula of Simons' type has been improved by K. Nomizu and B. Smyth [2] in 1969. Based on the new formula of Simons' type, they have determined hypersurfaces of non-negative sectional curvature and constant mean curvature immersed in the Euclidean space or the sphere under the additional assumption which is satisfied if M is compact. In the present paper, we first obtain a formula of Simons' type (2.11) in the case of a hypersurface M immersed with constant mean curvature in a conformally flat Riemannian manifold M^^〜 which satisfies some additional assumptions. These assumptions are described in terms of the Ricci operator of M^^〜 and naturally satisfied if M^^〜 is a space of constant sectional curvature. _Our main results are the determination of the immersions of M into M^^〜 when M admits non-negative sectional curvature.

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