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Let M be an n(n ≥ 3)-dimensional complete connected hypersurface in a unit sphere S(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− c2) × Sn−1(c), c = n−2 nr if r ≥ n−2 n−1 and S ≤… (More)
Let M be an n -dimensional compact Willmore Lagrangian submanifold in a complex projective space CPn and let S and H be the squared norm of the second fundamental form and the mean curvature of M . Denote by ρ2 = S−nH2 the non-negative function on M , K and Q the functions which assign to each point of M the infimum of the sectional curvature and Ricci… (More)
Let N p (c) be an (n+p)-dimensional connected Lorentzian space form of constant sectional curvature c and φ : M → N p (c) an n-dimensional spacelike submanifold in N p (c). The immersion φ : M → N p (c) is called a Willmore spacelike submanifold in N p (c) if it is a critical submanifold to the Willmore functional W (φ) = ∫
We study some Weingarten spacelike hypersurfaces in a de Sitter space S 1 (1). If the Weingarten spacelike hypersurfaces have two distinct principal curvatures, we obtain two classification theorems which give some characterization of the Riemannian product H(1−coth ̺)× S(1 − tanh ̺), 1 < k < n − 1 in S 1 (1), the hyperbolic cylinder H(1 − coth ̺) × S(1 −… (More)
In this paper, we would like to study space-like submanifolds in a de Sitter spaces S p (1). We define and discuss three Schrödinger operators LH , LR, LR/H and obtain some spectral characterizations of totally umbilical space-like submanifolds in terms of the first eigenvalue of the Schrödinger operators LH , LR and LR/H respectively.
In this paper, we investigate n-dimensional submanifolds with higher codimension in a unit sphere Sn+p(1). We obtain some rigidity results of submanifolds in Sn+p(1) with parallel mean curvature vector or with constant scalar curvature, which generalize some related rigidity results of hypersurfaces.
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)
In this paper, we study n(n ≥ 3)-dimensional complete connected and oriented space-like hypersurfaces Mn in an (n+1)-dimensional Lorentzian space form Mn+1 1 (c) with non-zero constant k-th (k < n) mean curvature and two distinct principal curvatures λ and μ. We give some characterizations of Riemannian product H(c1) ×Mn−m(c2) and show that the Riemannian… (More)