Abstract

Let M be a trans-Sasakian 3-manifold such that the Ricci curvature of the structure vector field vanishes. In this paper, it is proved that if M is compact, then it is homothetic to a cosymplectic manifold. Without the compactness assumption, we prove that M is locally isometric to the product of ℝ and a Kahler surface of constant curvature if the Ricci operator of M is of Codazzi type or cyclic parallel. We classify trans-Sasakian 3-manifolds with Killing type Ricci tensors. We construct some concrete examples to illustrate the above theorems.

中文翻译：

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具有类似爱因斯坦的 Ricci 算子的 Trans-Sasakian 3 流形

摘要

令M为跨 Sasakian 3 流形，使得结构矢量场的 Ricci 曲率消失。在这篇论文中，证明了如果M是紧致的，那么它与一个余辛流形是相似的。如果没有紧致性假设，我们证明如果M的 Ricci 算子是 Codazzi 类型或循环平行，则M与 ℝ 和恒定曲率 Kahler 曲面的乘积局部等距。我们使用 Killing 型 Ricci 张量对 trans-Sasakian 3 流形进行分类。我们构建了一些具体的例子来说明上述定理。