Submanifolds with parallel mean curvature vector in spheres

Submanifolds with parallel mean curvature vector in pinched Riemannian manifolds

“…For submanifolds with parallel mean curvature vector in arbitrary Riemannian manifolds, we obtain here the following integral inequality, which generalizes the inequalities obtained by other authors ( [1,10,11], etc. ): Theorem 1.13.…”

“…For submanifolds with parallel mean curvature vector in spheres, Walcy Santos extended the above theorem for higher codimensions [10]. Note that in the codimension one case, the mean curvature vector h is parallel if and only if H = |h| is constant.…”

“…The case of equality in Theorem (1.23) was inspired in the proof of the theorem of Santos [10]. , condition (1.24) is immediately satisfied and (1.25) becomes the hypothesis of Theorem (1.12).…”

Hypersurfaces in a sphere with constant mean curvature

“…M. Okumura [6,7] first discussed the general case and gave a pinching constant of S, but it is not sharp. Recently the sharp ones were obtained by H. Alencar-M. do Carmo [1] for p = 1, W. Santos [8] for p > 1 and H. W. Xu [11] for p ≥ 1 respectively. But all of them were expressed by the mean curvature H. S. T. Yau [12] obtained a pinching constant for p > 1 which depended only on n and p. H. W. Xu [10] improved Yau's result, but far from sharpness.…”

Spectral characterization and Schrödinger operator of space-like submanifolds

“…We need the following Lemma Lemma 2.1 ( [13]). Let A, B be symmetric n × n matrices satisfying AB = BA and trA = trB = 0.…”