On generalized $ξ$-Parallel Maps

Abstract

In this paper, we introduce and study the notion of generalized $ξ$-parallel maps between Riemannian manifolds, where $ξ$ is a vector field on the target manifold. We define the associated energy functional and derive its first variation formula, leading to the Euler--Lagrange equation characterizing generalized $ξ$-parallel maps. We establish fundamental properties of this new class of maps and investigate its relationship with harmonic and biharmonic maps. Furthermore, we study generalized $ξ$-parallel curves in space forms and examine generalized $ξ$-parallel submanifolds of space forms admitting concircular vector fields.