On the Interpolating Sesqui-Harmonicity of Vector Fields

TL;DR

This paper studies the conditions under which vector fields on Riemannian manifolds are interpolating sesqui-harmonic, providing characterization theorems and showing that under certain conditions, such vector fields must be parallel.

Contribution

It introduces the concept of interpolating sesqui-harmonic vector fields and maps, providing characterization theorems and extending results to Lie groups with compact quotients.

Findings

Interpolating sesqui-harmonic vector fields are characterized by specific conditions.

On compact, oriented manifolds, such vector fields are parallel.

Extension of results to left-invariant vector fields on Lie groups with compact quotients.

Abstract

This article deals with the interpolating sesqui-harmonicity of a vector field $X$ viewed as a map from a Riemannian manifold $(M,g)$ to its tangent bundle $TM$ endowed with the Sasaki metric $g_{S}$. We show characterization theorem for $X$ to be interpolating sesqui-harmonic map. We give also the critical point condition which characterizes interpolating sesqui-harmonic vector fields. When $(M,g)$ is compact and oriented and under some conditions, we prove that $X$ is an interpolating sesqui-harmonic vector field (resp. interpolating sesqui-harmonic map) if and only if $X$ is parallel. Moreover, we extend this result for a left-invariant vector field on a Lie group $G$ having a discrete subgroup $\Gamma$ such that the quotient $\Gamma\backslash G$ is compact.