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Let M be an n(n ≥ 3)-dimensional complete connected hyper-surface in a unit sphere S n+1 (1). In this paper, we show that (1) if M has non-zero mean curvature and constant scalar curvature n(n − 1)r and two distinct principal curvatures, one of which is simple, then M is isometric to the Riemannian product S 1 (√ 1 − c 2) × S n−1 (c), c 2 = n−2 nr if r ≥(More)
For a given (n − 1)-dimensional hypersurface x : M → R, consider the Laguerre form Φ, the Laguerre tensor L and the Laguerre second fundamental form B of the immersion x. In this article, we address the case when the Laguerre form of x is parallel, i.e., ∇Φ ≡ 0. We prove that ∇Φ ≡ 0 is equivalent to Φ ≡ 0, provided that either L+λB+μg = 0 for some smooth(More)
Let N n+p p (c) be an (n+p)-dimensional connected Lorentzian space form of constant sectional curvature c and φ : M → N n+p p (c) an n-dimensional spacelike submanifold in N n+p p (c). The immersion φ : M → N n+p p (c) is called a Willmore spacelike submanifold in N n+p p (c) if it is a critical submanifold to the Willmore functional W (φ) = ∫ M ρ n dv = ∫(More)