Abstract

Let (M³, g) be a three dimensional almost coKähler manifold such that the Reeb vector field ξ is an eigenvector field of the Ricci operator Q, i.e. Qξ = ρξ, where ρ is a smooth function on M. In this article, we prove that if g represents a Cotton soliton with potential vector field being collinear with ξ, or a gradient Cotton soliton, then M is coKähler or locally conformally flat. Furthermore, when g represents a nontrivial Cotton soliton with potential vector field being orthogonal to ξ, we prove that M is coKähler or locally isometric to one of the following Lie groups: E(2) or E(1, 1) if ρ is constant along ξ. Finally, for a (κ, µ, ν)-almost coKähler manifold, we also consider that g is a nontrivial Cotton soliton with potential vector field being orthogonal to ξ.

摘要

令 ( M ³ , g ) 是一个三维几乎 coKähler 流形，使得 Reeb 向量场ξ是 Ricci 算子的特征向量场问，即Qξ = ρξ，其中ρ是M上的平滑函数。在本文中，我们证明如果G表示势矢量场与ξ共线的Cotton 孤子，或梯度 Cotton 孤子，则M是 coKähler 或局部保形平坦。此外，当g表示具有与ξ正交的势向量场的非平凡 Cotton 孤子时，我们证明M是 coKähler 或与以下李群之一局部等距：E (2) 或E (1, 1) 如果ρ是常数沿ξ。最后，对于 ( κ, µ, ν )-几乎 coKähler 流形，我们还认为g是一个非平凡的棉花孤子，其势向量场与ξ。