The main goal of this paper is to study the properties of generalized Ricci recurrent perfect fluid spacetimes and the generalized Ricci recurrent (generalized Robertson–Walker (GRW)) spacetimes. It is proven that if the generalized Ricci recurrent perfect fluid spacetimes satisfy the Einstein’s field equations without cosmological constant, then the isotropic pressure and the energy density of the perfect fluid spacetime are invariant along the velocity vector field of the perfect fluid spacetime. In this series, we show that a generalized Ricci recurrent perfect fluid spacetime satisfying the Einstein’s field equations without cosmological constant is either Ricci recurrent or Ricci symmetric. An n-dimensional compact generalized Ricci recurrent GRW spacetime with almost Ricci soliton is geodesically complete, provided the soliton vector field of almost Ricci soliton is timelike. Also, we prove that a (GR)_n GRW spacetime is Einstein. The properties of (GR)_n GRW spacetimes equipped with almost Ricci soliton are studied.

中文翻译：

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广义 Ricci 循环时空和 GRW 时空

本文的主要目的是研究广义 Ricci 循环完美流体时空和广义 Ricci 循环（广义 Robertson-Walker (GRW)）时空的性质。证明了如果广义Ricci递归完美流体时空满足爱因斯坦场方程而没有宇宙学常数，则完美流体时空的各向同性压力和能量密度沿完美流体时空的速度矢量场是不变的。在本系列中，我们展示了满足爱因斯坦场方程且没有宇宙学常数的广义 Ricci 递归完美流体时空是 Ricci 递归或 Ricci 对称的。一个n几乎Ricci孤子的一维紧致广义Ricci递归GRW时空是测地完备的，前提是几乎Ricci孤子的孤子向量场是时间似的。此外，我们证明了 (GR)_nGRW时空是爱因斯坦。(GR) 的性质_n研究了几乎配备了里奇孤子的 GRW 时空。