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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)
We study some Weingarten spacelike hypersurfaces in a de Sitter space S n+1 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 k (1−coth 2 ̺)× S n−k (1 − tanh 2 ̺), 1 < k < n − 1 in S n+1 1 (1), the hyperbolic cylinder H 1… (More)
Let M be an n-dimensional compact Willmore Lagrangian submanifold in a complex projective space CP n and let S and H be the squared norm of the second fundamental form and the mean curvature of M. Denote by ρ 2 = S − nH 2 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)
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 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.
In this paper, we study the conformal geometry of conformal isoparametric spacelike hypersurfaces in conformal space Q n+1 1. We obtain the classification of the conformal isoparametric spacelike hypersurfaces in Q n+1 1 with three distinct conformal principal curvatures, one of which is simple, and the classification of the conformal isoparametric… (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)