A link between harmonicity of 2-distance functions and incompressibility of canonical vector fields

Abstract

Let $M$ be a Riemannian submanifold of a Riemannian manifold $\tilde M$ equipped with a concurrent vector field $\tilde Z$. Let $Z$ denote the restriction of $\tilde Z$ along $M$ and let $Z^T$ be the tangential component of $Z$ on $M$, called the canonical vector field of $M$. The 2-distance function $\delta^2_Z$ of $M$ (associated with $Z$) is defined by $\delta^2_Z=\$. In this article, we initiate the study of submanifolds $M$ of $\tilde M$ with incompressible canonical vector field $Z^T$ arisen from a concurrent vector field $\tilde Z$ on the ambient space $\tilde M$. First, we derive some necessary and sufficient conditions for such canonical vector fields to be incompressible. In particular, we prove that the 2-distance function $\delta^2_Z$ is harmonic if and only if the canonical vector field $Z^T$ on $M$ is an incompressible vector field. Then we provide some applications of our main results.