论文标题
地球轨道空间和相关伪里曼尼亚歧管的家族
Families of Geodesic Orbit Spaces and Related Pseudo-Riemannian Manifolds
论文作者
论文摘要
两个同质的伪riemannian流形$（g/h，ds^2）$和$（g'/h'，ds'^2）$属于相同{\ it real form family}，如果它们的复杂化$（g _ {\ mathbb c} $（g'_ {\ Mathbb C}/H'_ {\ Mathbb C}，DS'^2 _ {\ Mathbb C}）$是等值的。关键是，在许多情况下，特定空间$（g/h，ds^2）$具有有趣的属性，并且这些属性适用于其真实形式家族的空间。在这里，我们证明，如果$（g/h，ds^2）$是一个地理轨道空间，具有还原分解$ \ mathfrak {g} = \ mathfrak {h} + \ mathfrak {m Mathfrak {m} $，那么同一成员也占据了其真实形式的家族的所有成员。特别地，我们对紧凑的大地测量轨道riemannian歧管的理解提供了有关地球轨道伪里人歧管流形的信息。对于自然还原空间，通勤空间以及在大多数情况下，对于弱对称空间，我们还证明了相似的结果。我们以讨论这些真实形式家庭的包含，对D'Atri空间的讨论以及许多开放问题的讨论结束。
Two homogeneous pseudo-riemannian manifolds $(G/H, ds^2)$ and $(G'/H', ds'^2)$ belong to the same {\it real form family} if their complexifications $(G_{\mathbb C}/H_{\mathbb C}, ds_{\mathbb C}^2)$ and $(G'_{\mathbb C}/H'_{\mathbb C}, ds'^2_{\mathbb C})$ are isometric. The point is that in many cases a particular space $(G/H, ds^2)$ has interesting properties, and those properties hold for the spaces in its real form family. Here we prove that if $(G/H, ds^2)$ is a geodesic orbit space with a reductive decomposition $\mathfrak{g} = \mathfrak{h} + \mathfrak{m}$, then the same holds all the members of its real form family. In particular our understanding of compact geodesic orbit riemannian manifolds gives information on geodesic orbit pseudo-riemannian manifolds. We also prove similar results for naturally reductive spaces, for commutative spaces, and in most cases for weakly symmetric spaces. We end with a discussion of inclusions of these real form families, a discussion of D'Atri spaces, and a number of open problems.
