A sharp integral inequality for closed spacelike submanifolds immersed in the de Sitter space

Nonexistence and umbilicity of spacelike submanifolds with second fundamental form locally timelike

“…where α = P γ,q (sup M | |) > 0. Now, assuming that M n is noncompact, since (10), (11) and (13) were verified and M n has polynomial volume growth, we are able to apply Lemma 3.3 to obtain that | | 2 ≡ 0 and, consequently, M n is a totally umbilical submanifold.…”

“…Next, Chen, Liu, and Shu [10] obtained some integral inequalities of Simons type and rigidity theorems concerning compact spacelike submanifolds of S n+ p q . Afterwards, the first and second authors jointly with dos Santos [13] extended a technique developed by Alías and Meléndez [3] to prove a sharp integral inequality involving the norm of the total umbilicity tensor of a compact spacelike submanifold with constant scalar curvature immersed with parallel normalized mean curvature vector field in S n+ p p and they used it to characterize totally umbilical round spheres S n (r ) of S n+1 1 → S n+ p p . Motivated by these works, the purpose of the present paper is to investigate the nonexistence and umbilicity of n-dimensional (n ≥ 3) spacelike submanifolds immersed with parallel mean curvature vector in the (n + p)dimensional de Sitter space S n+ p q of index 1 ≤ q ≤ p. First, we show that there does not exist an ndimensional complete spacelike submanifold M n immersed with parallel mean curvature vector, whose the second fundamental form is locally timelike in S n+ p q and the mean curvature H satisfies 4(n − 1)…”