In this paper, we explore a set of geometric properties of Mixed Super Quasi-Einstein (MSQE) manifolds and provide examples of both Riemannian and Lorentzian MSQE manifolds to demonstrate their existence. Furthermore, we examine MSQE spacetimes in the context of the space-matter tensor, discussing several related properties. Finally, we establish the existence of an MSQE spacetime through a nontrivial example.

中文翻译：

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混合超准爱因斯坦流形的研究及其在广义相对论中的应用

在本文中，我们探索了混合超拟爱因斯坦 (MSQE) 流形的一组几何性质，并提供了黎曼 MSQE 流形和洛伦兹 MSQE 流形的示例来证明它们的存在。此外，我们在空间-物质张量的背景下研究 MSQE 时空，讨论几个相关的属性。最后，我们通过一个重要的例子证明了 MSQE 时空的存在性。