论文标题
在Lorentz歧管的几何轨道特性上
On the Geometric Orbit Property for Lorentz Manifolds
论文作者
论文摘要
已经对Riemannian歧管进行了深入研究地球轨道特性。地球轨道空间是均匀的，并使用等轴测组的Lie代数来简化许多结构性问题。弱对称的riemannian歧管是地球轨道空间。在这里，我们为伪里曼尼亚人的歧管定义了“自然还原”，并注意它们是地球轨道空间。几年前，两位作者证明了弱对称的伪riemannian流形是地球轨道空间。特别是这些结果适用于伪里曼尼亚洛伦兹歧管。我们的主要结果是定理4.2和5.1。在Riemannian病例中，Geodesic Orbit Nilmanifold的Nilpotent等轴测组是Abelian或$ 2 $ - 步骤Nilpotent。示例表明，在伪里曼尼亚案中，这会剧烈失败。在这里，我们专注于Lorentz Nilmanifolds $ g/h $的Geodesic Orbit物业，其中$ g = n \ rtimes h $和$ n $ nilpotent。当$ [\ mathfrak {n}，\ mathfrak {n}] $上的度量不超过时，Theorem 4.2表明，$ n $最多是$ 2 $ step nilpotent，如Riemannian情况，或者是$ 4 $ step nilpotent，但不能是$ 3 $ -Step nilpotent，不能是$ 3 $ -Step nililpotent。示例表明，这些界限是最好的。出乎意料的是，定理5.1表明，当$ [\ mathfrak {n}，\ mathfrak {n}] $变简时，$ n $最多是$ 2 $ - 步骤nilpotent。这两种定理都提供了其他结构信息，并专门研究自然还原和弱对称的Lorentz Nilmanifolds。 关键词：地球轨道空间； Lorentz Nilmanifold；弱对称空间；自然还原空间；伪里曼尼亚人歧管。
The geodesic orbit property has been studied intensively for Riemannian manifolds. Geodesic orbit spaces are homogeneous and allow simplifications of many structural questions using the Lie algebra of the isometry group. Weakly symmetric Riemannian manifolds are geodesic orbit spaces. Here we define "naturally reductive" for pseudo-Riemannian manifolds and note that they are geodesic orbit spaces. A few years ago two of the authors proved that weakly symmetric pseudo-Riemannian manifolds are geodesic orbit spaces. In particular these results apply to pseudo-Riemannian Lorentz manifolds. There our main results are Theorems 4.2 and 5.1. In the Riemannian case the nilpotent isometry group for a geodesic orbit nilmanifold is abelian or $2$-step nilpotent. Examples show that this fails dramatically in the pseudo-Riemannian case. Here we concentrate on the geodesic orbit property for Lorentz nilmanifolds $G/H$ with $G = N \rtimes H$ and $N$ nilpotent. When the metric is nondegenerate on $[\mathfrak{n},\mathfrak{n}]$, Theorem 4.2 shows that $N$ either is at most $2$-step nilpotent as in the Riemannian situation, or is $4$-step nilpotent, but cannot be $3$-step nilpotent. Examples show that these bounds are the best possible. Surprisingly, Theorem 5.1 shows that $N$ is at most $2$-step nilpotent when the metric is degenerate on $[\mathfrak{n},\mathfrak{n}]$. Both theorems give additional structural information and specialize to naturally reductive and to weakly symmetric Lorentz nilmanifolds. Key Words: Geodesic Orbit Space; Lorentz nilmanifold; Weakly Symmetric Space; Naturally Reductive Space; Pseudo-Riemannian Manifold.
