relating diameter and mean curvature for submanifolds of euclidean space

Umbilicity of (Space-Like) Submanifolds of Pseudo-Riemannian Space Forms

We study umbilic (space-like) submanifolds of pseudo-Riemannian space forms, then define totally semi-umbilic space-like submanifold of pseudo Euclidean space and relate this notion to umbilicity. Finally we give characterization of total semi-umbilicity for space-like submanifolds contained in pseudo sphere or pseudo hyperbolic space or the light cone.A pseudo-Riemannian submanifold M in (a...

RICCI CURVATURE OF SUBMANIFOLDS OF A SASAKIAN SPACE FORM

Involving the Ricci curvature and the squared mean curvature, we obtain basic inequalities for different kind of submaniforlds of a Sasakian space form tangent to the structure vector field of the ambient manifold. Contrary to already known results, we find a different necessary and sufficient condition for the equality for Ricci curvature of C-totally real submanifolds of a Sasakian space form...

$L_1$-Biharmonic Hypersurfaces in Euclidean Spaces with Three Distinct Principal Curvatures

Chen's biharmonic conjecture is well-known and stays open: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. In this paper, we consider an advanced version of the conjecture, replacing $Delta$ by its extension, $L_1$-operator ($L_1$-conjecture). The $L_1$-conjecture states that any $L_1$-biharmonic Euclidean hypersurface is 1-minimal. We prove that the $L_1$-conje...

Problems in the Geometry of Submanifolds

This article grew out of several talks that the author presented at the Banach Institute and at the University of Bialystok in Poland during November of 2001. It describes six problems from the geometry of submanifolds. Some of the problems come from the theory of constant curvature submanifolds in Euclidean space, as well as applications of Morse theory of the height function to the problem of...

Shape Operator of Slant Submanifolds in Kenmotsu Space Forms