Аналитическое продолжение локально однородного и неоднородного риманова многообразия

Abstract

Let’s consider the Lie algebra 𝔤 of all Killing vector fields of a Riemannian analytic manifold, its stationary subalgebra 𝔥, the simply connected Lie group 𝐺 corresponding to the Lie algebra 𝔤 and the subgroup 𝐻 corresponding to the Lie subalgebra 𝔥. The set of left adjacent classes 𝐺𝐻⁄ forms a homogeneous manifold if and only If 𝐻 is closed in 𝐺. We study the properties of the Lie algebra 𝔤 and its subalgebra 𝔥 under which 𝐻 is closed in 𝐺. The following category of Riemannian analytic manifolds is also studied. The objects of this category are oriented Riemannian analytic manifolds having open subsets isometric to each other and, consequently, the same algebra 𝔤 of Killing vector fields. It is assumed that the algebra 𝔤 has no center. Morphisms of this category are locally isometric maps 𝑓:𝑀⟶𝑁 preserving orientation and Killing vector fields. Moreover, the maps 𝑓 are defined on the entire manifold 𝑀 with the exception of the set 𝑆 of codimension at least two, consisting of fixed points of orientation-preserving isometries between open subsets of the manifold 𝑀. This category has a univer-sally attractive object. This is a so-called quasi-complete manifold, which by definition is unextendable manifold that does not admit nontrivial orientation-preserving and vector Killing fields isometries between its open subsets. For an arbitrary Riemannian analytic metric, a pseudo-field Riemannian analytic manifold is defined. This is a simply connected manifold 𝑀 for which there is no locally isometric map 𝑓:𝑀⟶𝑁 define on the whole 𝑀 and preserving orientation and killing vector fields. Where 𝑁 is a simply connected Riemannian analytic manifold other than 𝑀.