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y − x ≥ v(y) − v(x) d − h ≥ b + |M n | 2 − 1 a − 1 ≥ b 2 − 1 a − 1 in this case as well. Again, the unboundedness of D(v, M) follows.
We completely classify constant mean curvature hypersurfaces (CMC) with constant δ-invariant in the unit 4-sphere S 4 and in the Euclidean 4-space E 4 .
We present a method to construct a large family of Lagrangian surfaces in complex Euclidean plane C 2 by using Legendre curves in the 3-sphere and in the anti de Sitter 3-space or, equivalently, by using spherical and hyperbolic curves, respectively. Among this family, we characterize minimal, constant mean curvature, Hamiltonian-minimal and Willmore… (More)
A new geometric invariant will be introduced, studied and determined on compact symmetric spaces. Introduction. We will introduce a new invariant on Riemannian manifolds, which is especially interesting on compact symmetric spaces, and we will determine the invariant for the compact symmetric spaces, thus amplifying the announcement
Lagrangian //-umbilical submanifolds are the "simplest" Lagrangian submanifolds next to totally geodesic ones in complex-space-forms. The class of Lagrangian //-umbilical submanifolds in complex Euclidean spaces includes Whitney's spheres and Lagrangian pseudo-spheres. For each submanifold M of Euclidean «-space and each unit speed curve F in the complex… (More)
Einstein manifolds are trivial examples of gradient Ricci soli-tons with constant potential function and thus they are called trivial Ricci solitons. In this paper, we prove two characterizations of compact shrinking trivial Ricci solitons.
A slant immersion was introduced in  as an isometric immersion of a Rie-mannian manifold into an almost Hermitian manifold (˜ M , g, J) with constant Wirtinger angle. From J-action point of view, the most natural surfaces in an almost Hermitian manifold are slant surfaces. Flat slant surfaces in complex space forms have been studied in [3, 4]. In this… (More)
By applying the spectral decomposition of a submanifold of a Euclidean space, we derive several sharp geometric inequalities which provide us some best possible relations between volume, center of mass, circumscribed radius, inscribed radius, order, and mean curvature of the submanifold. Several of our results sharpen some well-known geometric inequalities.