ABSTRACT. In this paper, We prove that every (e)-sasakian manifold is a hypersurface of an indefinite kaehlerian manifold, and give a necessary and sufficient condition for a Riemannian manifold to be an (e)-sasakian manifold.KEY WORDS AND PHRASES: ()-sasakian manifolds; real hypersurface; indefinite kaehlerian manifolds; (e)-almost contact structure.
A classical nonlocal curvature flow preserving the enclosed area is reinvestigated. The uniform upper bound and lower bound of curvature are established for the first time. As a result, a detailed proof is presented for the asymptotic behavior of the flow.
Abstract. In this note, we generalize the weak maximum principle in [4] to the case of complete linear Weingarten hypersurface in Riemannian space form M n+1 (c) (c = 1, 0, −1), and apply it to estimate the norm of the total umbilicity tensor. Furthermore, we will study the linear Weingarten hypersurface in S n+1 (1) with the aid of this weak maximum principle and extend the rigidity results in Li, Suh, Wei [13] and Shu [15] to the case of complete hypersurface.
In this paper, by modifying Cheng-Yau's technique to complete spacelike submanifolds in Q nþp p ðcÞ, we prove a rigidity theorem for complete spacelike submanifolds in the de Sitter space with parallel normalized mean curvature vector. As a corollary, we have the Corollary 1.1 of [7].
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