H-Convex Riemannian Submanifolds

TL;DR

This paper generalizes the concept of convexity to Riemannian submanifolds of arbitrary codimension by using the mean curvature vector, extending prior work on convex hypersurfaces and exploring their properties.

Contribution

It introduces the notion of H-convexity for Riemannian submanifolds of any codimension, replacing the normal vector with the mean curvature vector, and provides a characterization of strict H-convexity.

Findings

Established the definition of H-convexity for submanifolds of arbitrary codimension.

Derived a characterization of strict H-convexity based on mean curvature vector.

Connected H-convexity with existing concepts of convexity and curvature in Riemannian geometry.

Abstract

Having in mind the well known model of Euclidean convex hypersurfaces [4], [5], and the ideas in [1] many authors defined and investigate convex hypersurfaces of a Riemannian manifold. As it was proved by the first author in [7], there follows the interdependence between convexity and Gauss curvature of the hypersurface. In this paper we define $H$-convexity of a Riemannian submanifold of arbitrary codimension, replacing the normal versor of a hypersurface with the mean curvature vector $H$. A characterization, used by B.Y. Chen [2], [3] as the definition of strictly $H$-convexity, it is obtained.