Аналитическое продолжение римановых аналитических многообразий малой размерности

Abstract

Abstract. An analytic extension of a locally given Riemannian analytic metric to a non-extendable Riemannian analytic manifold is considered. There are an infinite number of such extensions, and most of these extensions are very unnatural. The search for the most natural extensions leads to a generalization of the concept of completeness of a Riemannian manifold. It is possible to define a so called compressed manifold for metrics whose Lie algebra of Killing vector fields has no center. It is a universally attracting object in the category of all locally isometric Riemannian analytic manifolds. Morphisms of this category are locally isometric mappings 𝑓: 𝑀\𝑆, where 𝑆 is the set of fixed points of all local isometries of 𝑀 into itself that preserve orientation and Killing vector fields. For an arbitrary class of locally isometric Riemannian analytic manifolds, a definition of a pseudocomplete manifold is given. In contrast to a contracted manifold, a pseudocomplete manifold is complete if a complete manifold exists in the given class. A Riemannian analytic simply connected manifold M is called pseudocomplete if it has the following properties. M is non-extendable. There is no locally isometric covering mapping 𝑓: 𝑀 → 𝑁 where 𝑁 is a simply connected Riemannian analytic manifold, 𝑓(𝑀) is an open subset of 𝑁 not equal to 𝑁. In contrast to contracted manifolds, a pseudocomplete manifold is not unique in the class of locally isometric Riemannian analytic manifolds. Among the pseudocomplete manifolds, the most compressed regular pseudocomplete manifolds are defined. A classification of pseudocomplete manifolds of dimension 2 and 3 is given.