AI 中文总结
研究黎曼流形间广义ξ-平行映射，定义相关能量泛函并推导变分公式得欧拉-拉格朗日方程，建立其基本性质，探讨与调和及双调和映射关系，还研究了空间形式中的广义ξ-平行曲线及子流形。
AI 中文摘要
本文引入并研究了黎曼流形间广义ξ-平行映射的概念，其中ξ是目标流形上的向量场。定义了相关能量泛函并推导其一阶变分公式，得出刻画广义ξ-平行映射的欧拉-拉格朗日方程。建立了这类新映射的基本性质，研究其与调和及双调和映射的关系。还研究了空间形式中的广义ξ-平行曲线及允许共圆向量场的空间形式的广义ξ-平行子流形。
英文摘要
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.
