In this paper we study Riemannian manifolds \((M^n, g)\) admitting an m-quasi-Einstein metric with V as its potential vector field. We derive an integral formula for compact m-quasi-Einstein manifolds and prove that the vector field V vanishes under certain integral inequality. Next, we prove that if the metrically equivalent 1-form \(V^{\flat }\) associated with the potential vector field is a harmonic 1-form, then V is an infinitesimal harmonic transformation. Moreover, if M is compact then it is Einstein. Some more results were obtained when (i) V generates an infinitesimal harmonic transformation, (ii) V is a conformal vector field.

中文翻译：

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满足势矢量场某些条件的m-拟爱因斯坦度量

在本文中，我们研究了黎曼流形\（（M ^ n，g）\）接受m-准爱因斯坦度量，其中V是其潜在的矢量场。我们推导了紧凑的m-拟爱因斯坦流形的积分公式，并证明矢量场V在某些积分不等式下消失。接下来，我们证明如果与势矢量场相关的度量等效的1形式\（V ^ {\ flat} \）是谐波1形式，则V是无穷小谐波变换。而且，如果M是紧凑的，那么它就是爱因斯坦。当（i）V生成无限次谐波变换，（ii）V是共形矢量场。