带有反循环结构的三维近爱因斯坦流形

Three-Dimensional Almost Einstein Manifolds with Skew-Circulant Structures

摘要: 三维黎曼流形上添加一个局部分量为Toeplitz矩阵的循环结构，可以应用于线性编码、图论、震动分析和调和伯格曼空间的研究等。而带有反循环结构的黎曼流形与爱因斯坦流形有着密切联系。本文从反循环角度出发，研究带有反循环结构的三维黎曼流形，给出反循环结构相容的等价条件。并利用反循环结构构建新的度量，结合新的度量证明这种流形是近爱因斯坦流形，进而给出近爱因斯坦流形的一些曲率特性。最后给出反循环结构作用到李群上的例子。

Abstract: An imposing circulant structure with local components of Toeplitz matrices on three-dimensional Riemannian manifolds can be applied to the linear coding, graph theory, vibration analysis and the study of harmonic Bergman space. The Riemannian manifold with skew-circulant structure whose local components are skew-circulant matrices is closely related to Einstein manifold. In this paper, from the point of view of skew-circulant structure, the equivalent conditions compatible with the skew-circulant structure are obtained, and some new metrics are constructed by using the skew-circulant structure. Combined with the new metric, it is showed that this kind of manifold with skew-circulant structure is an almost Einstein manifold, and then some curvature properties of the almost Einstein manifold are derived. Finally, an example of skew-circulant structure acting on Lie groups is given.