Almost Kenmotsu manifolds and local symmetry
Almost Kenmotsu manifolds and local symmetry
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DOI：
10.36045/bbms/1179839227
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发表时间：
2007-06
影响因子：
0.5
通讯作者：
G. Dileo;A. Pastore
中科院分区：
文献类型：
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作者：
G. Dileo;A. Pastore
我们考虑局部对称的几乎Kenmotsu流形，证明了这样的流形是Kenmotsu流形的充要条件是该流形关于Reeb向量场ξ的Lie导数为零。进一步，假设对于一个(2n+1)维局部对称的几乎Kenmotsu流形，这样的Lie导数不为零，且对任何X，Y都满足Rxyξ=0，我们证明了该流形局部等距于(n+1)维常曲率流形ξ4和n维平坦流形的黎曼乘积.我们给出了这样一个流形的例子。介绍了可微流形M上的一个几乎接触结构，它由一个(1，1)型张量场φ、一个向量场ξ和一个满足ηi+φ=−和η⊗ξ)=1的1-形式η给出，这意味着φ(ξ)=0和η◦φ=0.此外，在乘积流形M×R上，可以由J定义一个几乎复数结构J(
We consider locally symmetric almost Kenmotsu manifolds showing that such a manifold is a Kenmotsu manifold if and only if the Lie derivative of the structure, with respect to the Reeb vector field ξ, vanishes. Furthermore, assuming that for a (2n + 1)-dimensional locally symmetric almost Kenmotsu manifold such Lie derivative does not vanish and the curvature satisfies RXY ξ = 0 for any X,Y orthogonal to ξ, we prove that the manifold is locally isometric to the Riemannian product of an (n+1)-dimensional manifold of constant curvature −4 and a flat n-dimensional manifold. We give an example of such a manifold. Introduction An almost contact structure on a differentiable manifold M is given by a tensor field φ of type (1, 1), a vector field ξ and a 1-form η satisfying φ = − I + η⊗ ξ and η(ξ) = 1, which imply that φ(ξ) = 0 and η ◦ φ = 0. Furthermore, on the product manifold M × R one can define an almost complex structure J by J (